# Fraction reduction

Fraction reduction
You are encouraged to solve this task according to the task description, using any language you may know.
```              There is a fine line between numerator and denominator.       ─── anonymous
```

A method to   "reduce"   some reducible fractions is to   cross out   a digit from the numerator and the denominator.   An example is:

```       16                                                  16
────     and then (simply) cross─out the sixes:      ────
64                                                  64
```

resulting in:

```        1
───
4
```

Naturally,   this "method" of reduction must reduce to the proper value   (shown as a fraction).

This "method" is also known as   anomalous cancellation   and also   accidental cancellation.

(Of course,   this "method" shouldn't be taught to impressionable or gullible minds.)       😇

Find and show some fractions that can be reduced by the above "method".

•   show 2-digit fractions found   (like the example shown above)
•   show 3-digit fractions
•   show 4-digit fractions
•   show 5-digit fractions   (and higher)       (optional)
•   show each (above) n-digit fractions separately from other different n-sized fractions, don't mix different "sizes" together
•   for each "size" fraction,   only show a dozen examples   (the 1st twelve found)
•   (it's recognized that not every programming solution will have the same generation algorithm)
•   for each "size" fraction:
•   show a count of how many reducible fractions were found.   The example (above) is size 2
•   show a count of which digits were crossed out   (one line for each different digit)
•   for each "size" fraction,   show a count of how many were found.   The example (above) is size 2
•   show each n-digit example   (to be shown on one line):
•   show each n-digit fraction
•   show each reduced n-digit fraction
•   show what digit was crossed out for the numerator and the denominator

•   only proper fractions and their reductions   (the result)   are to be used   (no vulgar fractions)
•   only positive fractions are to be used   (no negative signs anywhere)
•   only base ten integers are to be used for the numerator and denominator
•   no zeros   (decimal digit)   can be used within the numerator or the denominator
•   the numerator and denominator should be composed of the same number of digits
•   no digit can be repeated in the numerator
•   no digit can be repeated in the denominator
•   (naturally)   there should be a shared decimal digit in the numerator   and   the denominator
•   fractions can be shown as   16/64   (for example)

References

## C

Translation of: C#
`#include <stdbool.h>#include <stdio.h>#include <stdlib.h>#include <string.h> typedef struct IntArray_t {    int *ptr;    size_t length;} IntArray; IntArray make(size_t size) {    IntArray temp;    temp.ptr = calloc(size, sizeof(int));    temp.length = size;    return temp;} void destroy(IntArray *ia) {    if (ia->ptr != NULL) {        free(ia->ptr);         ia->ptr = NULL;        ia->length = 0;    }} void zeroFill(IntArray dst) {    memset(dst.ptr, 0, dst.length * sizeof(int));} int indexOf(const int n, const IntArray ia) {    size_t i;    for (i = 0; i < ia.length; i++) {        if (ia.ptr[i] == n) {            return i;        }    }    return -1;} bool getDigits(int n, int le, IntArray digits) {    while (n > 0) {        int r = n % 10;        if (r == 0 || indexOf(r, digits) >= 0) {            return false;        }        le--;        digits.ptr[le] = r;        n /= 10;    }    return true;} int removeDigit(IntArray digits, size_t le, size_t idx) {    static const int POWS[] = { 1, 10, 100, 1000, 10000 };    int sum = 0;    int pow = POWS[le - 2];    size_t i;    for (i = 0; i < le; i++) {        if (i == idx) continue;        sum += digits.ptr[i] * pow;        pow /= 10;    }    return sum;} int main() {    int lims = { { 12, 97 }, { 123, 986 }, { 1234, 9875 }, { 12345, 98764 } };    int count = { 0 };    int omitted = { {0} };    size_t upperBound = sizeof(lims) / sizeof(lims);    size_t i;     for (i = 0; i < upperBound; i++) {        IntArray nDigits = make(i + 2);        IntArray dDigits = make(i + 2);        int n;         for (n = lims[i]; n <= lims[i]; n++) {            int d;            bool nOk;             zeroFill(nDigits);            nOk = getDigits(n, i + 2, nDigits);            if (!nOk) {                continue;            }            for (d = n + 1; d <= lims[i] + 1; d++) {                size_t nix;                bool dOk;                 zeroFill(dDigits);                dOk = getDigits(d, i + 2, dDigits);                if (!dOk) {                    continue;                }                for (nix = 0; nix < nDigits.length; nix++) {                    int digit = nDigits.ptr[nix];                    int dix = indexOf(digit, dDigits);                    if (dix >= 0) {                        int rn = removeDigit(nDigits, i + 2, nix);                        int rd = removeDigit(dDigits, i + 2, dix);                        if ((double)n / d == (double)rn / rd) {                            count[i]++;                            omitted[i][digit]++;                            if (count[i] <= 12) {                                printf("%d/%d = %d/%d by omitting %d's\n", n, d, rn, rd, digit);                            }                        }                    }                }            }        }         printf("\n");         destroy(&nDigits);        destroy(&dDigits);    }     for (i = 2; i <= 5; i++) {        int j;         printf("There are %d %d-digit fractions of which:\n", count[i - 2], i);         for (j = 1; j <= 9; j++) {            if (omitted[i - 2][j] == 0) {                continue;            }            printf("%6d have %d's omitted\n", omitted[i - 2][j], j);        }         printf("\n");    }     return 0;}`
Output:
```16/64 = 1/4 by omitting 6's
19/95 = 1/5 by omitting 9's
26/65 = 2/5 by omitting 6's
49/98 = 4/8 by omitting 9's

132/231 = 12/21 by omitting 3's
134/536 = 14/56 by omitting 3's
134/938 = 14/98 by omitting 3's
136/238 = 16/28 by omitting 3's
138/345 = 18/45 by omitting 3's
139/695 = 13/65 by omitting 9's
143/341 = 13/31 by omitting 4's
146/365 = 14/35 by omitting 6's
149/298 = 14/28 by omitting 9's
149/596 = 14/56 by omitting 9's
149/894 = 14/84 by omitting 9's
154/253 = 14/23 by omitting 5's

1234/4936 = 124/496 by omitting 3's
1239/6195 = 123/615 by omitting 9's
1246/3649 = 126/369 by omitting 4's
1249/2498 = 124/248 by omitting 9's
1259/6295 = 125/625 by omitting 9's
1279/6395 = 127/635 by omitting 9's
1283/5132 = 128/512 by omitting 3's
1297/2594 = 127/254 by omitting 9's
1297/3891 = 127/381 by omitting 9's
1298/2596 = 128/256 by omitting 9's
1298/3894 = 128/384 by omitting 9's
1298/5192 = 128/512 by omitting 9's

12349/24698 = 1234/2468 by omitting 9's
12356/67958 = 1236/6798 by omitting 5's
12358/14362 = 1258/1462 by omitting 3's
12358/15364 = 1258/1564 by omitting 3's
12358/17368 = 1258/1768 by omitting 3's
12358/19372 = 1258/1972 by omitting 3's
12358/21376 = 1258/2176 by omitting 3's
12358/25384 = 1258/2584 by omitting 3's
12359/61795 = 1235/6175 by omitting 9's
12364/32596 = 1364/3596 by omitting 2's
12379/61895 = 1237/6185 by omitting 9's
12386/32654 = 1386/3654 by omitting 2's

There are 4 2-digit fractions of which:
2 have 6's omitted
2 have 9's omitted

There are 122 3-digit fractions of which:
9 have 3's omitted
1 have 4's omitted
6 have 5's omitted
15 have 6's omitted
16 have 7's omitted
15 have 8's omitted
60 have 9's omitted

There are 660 4-digit fractions of which:
14 have 1's omitted
25 have 2's omitted
92 have 3's omitted
14 have 4's omitted
29 have 5's omitted
63 have 6's omitted
16 have 7's omitted
17 have 8's omitted
390 have 9's omitted

There are 5087 5-digit fractions of which:
75 have 1's omitted
40 have 2's omitted
376 have 3's omitted
78 have 4's omitted
209 have 5's omitted
379 have 6's omitted
591 have 7's omitted
351 have 8's omitted
2988 have 9's omitted```

## C#

Translation of: Kotlin
`using System; namespace FractionReduction {    class Program {        static int IndexOf(int n, int[] s) {            for (int i = 0; i < s.Length; i++) {                if (s[i] == n) {                    return i;                }            }            return -1;        }         static bool GetDigits(int n, int le, int[] digits) {            while (n > 0) {                var r = n % 10;                if (r == 0 || IndexOf(r, digits) >= 0) {                    return false;                }                le--;                digits[le] = r;                n /= 10;            }            return true;        }         static int RemoveDigit(int[] digits, int le, int idx) {            int[] pows = { 1, 10, 100, 1000, 10000 };             var sum = 0;            var pow = pows[le - 2];            for (int i = 0; i < le; i++) {                if (i == idx) continue;                sum += digits[i] * pow;                pow /= 10;             }            return sum;        }         static void Main() {            var lims = new int[,] { { 12, 97 }, { 123, 986 }, { 1234, 9875 }, { 12345, 98764 } };            var count = new int;            var omitted = new int[5, 10];            var upperBound = lims.GetLength(0);            for (int i = 0; i < upperBound; i++) {                var nDigits = new int[i + 2];                var dDigits = new int[i + 2];                var blank = new int[i + 2];                for (int n = lims[i, 0]; n <= lims[i, 1]; n++) {                    blank.CopyTo(nDigits, 0);                    var nOk = GetDigits(n, i + 2, nDigits);                    if (!nOk) {                        continue;                    }                    for (int d = n + 1; d <= lims[i, 1] + 1; d++) {                        blank.CopyTo(dDigits, 0);                        var dOk = GetDigits(d, i + 2, dDigits);                        if (!dOk) {                            continue;                        }                        for (int nix = 0; nix < nDigits.Length; nix++) {                            var digit = nDigits[nix];                            var dix = IndexOf(digit, dDigits);                            if (dix >= 0) {                                var rn = RemoveDigit(nDigits, i + 2, nix);                                var rd = RemoveDigit(dDigits, i + 2, dix);                                if ((double)n / d == (double)rn / rd) {                                    count[i]++;                                    omitted[i, digit]++;                                    if (count[i] <= 12) {                                        Console.WriteLine("{0}/{1} = {2}/{3} by omitting {4}'s", n, d, rn, rd, digit);                                    }                                }                            }                        }                    }                }                Console.WriteLine();            }             for (int i = 2; i <= 5; i++) {                Console.WriteLine("There are {0} {1}-digit fractions of which:", count[i - 2], i);                for (int j = 1; j <= 9; j++) {                    if (omitted[i - 2, j] == 0) {                        continue;                    }                    Console.WriteLine("{0,6} have {1}'s omitted", omitted[i - 2, j], j);                }                Console.WriteLine();            }        }    }}`
Output:
```16/64 = 1/4 by omitting 6's
19/95 = 1/5 by omitting 9's
26/65 = 2/5 by omitting 6's
49/98 = 4/8 by omitting 9's

132/231 = 12/21 by omitting 3's
134/536 = 14/56 by omitting 3's
134/938 = 14/98 by omitting 3's
136/238 = 16/28 by omitting 3's
138/345 = 18/45 by omitting 3's
139/695 = 13/65 by omitting 9's
143/341 = 13/31 by omitting 4's
146/365 = 14/35 by omitting 6's
149/298 = 14/28 by omitting 9's
149/596 = 14/56 by omitting 9's
149/894 = 14/84 by omitting 9's
154/253 = 14/23 by omitting 5's

1234/4936 = 124/496 by omitting 3's
1239/6195 = 123/615 by omitting 9's
1246/3649 = 126/369 by omitting 4's
1249/2498 = 124/248 by omitting 9's
1259/6295 = 125/625 by omitting 9's
1279/6395 = 127/635 by omitting 9's
1283/5132 = 128/512 by omitting 3's
1297/2594 = 127/254 by omitting 9's
1297/3891 = 127/381 by omitting 9's
1298/2596 = 128/256 by omitting 9's
1298/3894 = 128/384 by omitting 9's
1298/5192 = 128/512 by omitting 9's

12349/24698 = 1234/2468 by omitting 9's
12356/67958 = 1236/6798 by omitting 5's
12358/14362 = 1258/1462 by omitting 3's
12358/15364 = 1258/1564 by omitting 3's
12358/17368 = 1258/1768 by omitting 3's
12358/19372 = 1258/1972 by omitting 3's
12358/21376 = 1258/2176 by omitting 3's
12358/25384 = 1258/2584 by omitting 3's
12359/61795 = 1235/6175 by omitting 9's
12364/32596 = 1364/3596 by omitting 2's
12379/61895 = 1237/6185 by omitting 9's
12386/32654 = 1386/3654 by omitting 2's

There are 4 2-digit fractions of which:
2 have 6's omitted
2 have 9's omitted

There are 122 3-digit fractions of which:
9 have 3's omitted
1 have 4's omitted
6 have 5's omitted
15 have 6's omitted
16 have 7's omitted
15 have 8's omitted
60 have 9's omitted

There are 660 4-digit fractions of which:
14 have 1's omitted
25 have 2's omitted
92 have 3's omitted
14 have 4's omitted
29 have 5's omitted
63 have 6's omitted
16 have 7's omitted
17 have 8's omitted
390 have 9's omitted

There are 5087 5-digit fractions of which:
75 have 1's omitted
40 have 2's omitted
376 have 3's omitted
78 have 4's omitted
209 have 5's omitted
379 have 6's omitted
591 have 7's omitted
351 have 8's omitted
2988 have 9's omitted```

## C++

Translation of: D
`#include <array>#include <iomanip>#include <iostream>#include <vector> int indexOf(const std::vector<int> &haystack, int needle) {    auto it = haystack.cbegin();    auto end = haystack.cend();    int idx = 0;    for (; it != end; it = std::next(it)) {        if (*it == needle) {            return idx;        }        idx++;    }    return -1;} bool getDigits(int n, int le, std::vector<int> &digits) {    while (n > 0) {        auto r = n % 10;        if (r == 0 || indexOf(digits, r) >= 0) {            return false;        }        le--;        digits[le] = r;        n /= 10;    }    return true;} int removeDigit(const std::vector<int> &digits, int le, int idx) {    static std::array<int, 5> pows = { 1, 10, 100, 1000, 10000 };     int sum = 0;    auto pow = pows[le - 2];    for (int i = 0; i < le; i++) {        if (i == idx) continue;        sum += digits[i] * pow;        pow /= 10;    }    return sum;} int main() {    std::vector<std::pair<int, int>> lims = { {12, 97}, {123, 986}, {1234, 9875}, {12345, 98764} };    std::array<int, 5> count;    std::array<std::array<int, 10>, 5> omitted;     std::fill(count.begin(), count.end(), 0);    std::for_each(omitted.begin(), omitted.end(),        [](auto &a) {            std::fill(a.begin(), a.end(), 0);        }    );     for (size_t i = 0; i < lims.size(); i++) {        std::vector<int> nDigits(i + 2);        std::vector<int> dDigits(i + 2);         for (int n = lims[i].first; n <= lims[i].second; n++) {            std::fill(nDigits.begin(), nDigits.end(), 0);            bool nOk = getDigits(n, i + 2, nDigits);            if (!nOk) {                continue;            }            for (int d = n + 1; d <= lims[i].second + 1; d++) {                std::fill(dDigits.begin(), dDigits.end(), 0);                bool dOk = getDigits(d, i + 2, dDigits);                if (!dOk) {                    continue;                }                for (size_t nix = 0; nix < nDigits.size(); nix++) {                    auto digit = nDigits[nix];                    auto dix = indexOf(dDigits, digit);                    if (dix >= 0) {                        auto rn = removeDigit(nDigits, i + 2, nix);                        auto rd = removeDigit(dDigits, i + 2, dix);                        if ((double)n / d == (double)rn / rd) {                            count[i]++;                            omitted[i][digit]++;                            if (count[i] <= 12) {                                std::cout << n << '/' << d << " = " << rn << '/' << rd << " by omitting " << digit << "'s\n";                            }                        }                    }                }            }        }         std::cout << '\n';    }     for (int i = 2; i <= 5; i++) {        std::cout << "There are " << count[i - 2] << ' ' << i << "-digit fractions of which:\n";        for (int j = 1; j <= 9; j++) {            if (omitted[i - 2][j] == 0) {                continue;            }            std::cout << std::setw(6) << omitted[i - 2][j] << " have " << j << "'s omitted\n";        }        std::cout << '\n';    }     return 0;}`
Output:
```16/64 = 1/4 by omitting 6's
19/95 = 1/5 by omitting 9's
26/65 = 2/5 by omitting 6's
49/98 = 4/8 by omitting 9's

132/231 = 12/21 by omitting 3's
134/536 = 14/56 by omitting 3's
134/938 = 14/98 by omitting 3's
136/238 = 16/28 by omitting 3's
138/345 = 18/45 by omitting 3's
139/695 = 13/65 by omitting 9's
143/341 = 13/31 by omitting 4's
146/365 = 14/35 by omitting 6's
149/298 = 14/28 by omitting 9's
149/596 = 14/56 by omitting 9's
149/894 = 14/84 by omitting 9's
154/253 = 14/23 by omitting 5's

1234/4936 = 124/496 by omitting 3's
1239/6195 = 123/615 by omitting 9's
1246/3649 = 126/369 by omitting 4's
1249/2498 = 124/248 by omitting 9's
1259/6295 = 125/625 by omitting 9's
1279/6395 = 127/635 by omitting 9's
1283/5132 = 128/512 by omitting 3's
1297/2594 = 127/254 by omitting 9's
1297/3891 = 127/381 by omitting 9's
1298/2596 = 128/256 by omitting 9's
1298/3894 = 128/384 by omitting 9's
1298/5192 = 128/512 by omitting 9's

12349/24698 = 1234/2468 by omitting 9's
12356/67958 = 1236/6798 by omitting 5's
12358/14362 = 1258/1462 by omitting 3's
12358/15364 = 1258/1564 by omitting 3's
12358/17368 = 1258/1768 by omitting 3's
12358/19372 = 1258/1972 by omitting 3's
12358/21376 = 1258/2176 by omitting 3's
12358/25384 = 1258/2584 by omitting 3's
12359/61795 = 1235/6175 by omitting 9's
12364/32596 = 1364/3596 by omitting 2's
12379/61895 = 1237/6185 by omitting 9's
12386/32654 = 1386/3654 by omitting 2's

There are 4 2-digit fractions of which:
2 have 6's omitted
2 have 9's omitted

There are 122 3-digit fractions of which:
9 have 3's omitted
1 have 4's omitted
6 have 5's omitted
15 have 6's omitted
16 have 7's omitted
15 have 8's omitted
60 have 9's omitted

There are 660 4-digit fractions of which:
14 have 1's omitted
25 have 2's omitted
92 have 3's omitted
14 have 4's omitted
29 have 5's omitted
63 have 6's omitted
16 have 7's omitted
17 have 8's omitted
390 have 9's omitted

There are 5087 5-digit fractions of which:
75 have 1's omitted
40 have 2's omitted
376 have 3's omitted
78 have 4's omitted
209 have 5's omitted
379 have 6's omitted
591 have 7's omitted
351 have 8's omitted
2988 have 9's omitted```

## D

Translation of: C#
`import std.range;import std.stdio; int indexOf(Range, Element)(Range haystack, scope Element needle)if (isInputRange!Range) {    int idx;    foreach (straw; haystack) {        if (straw == needle) {            return idx;        }        idx++;    }    return -1;} bool getDigits(int n, int le, int[] digits) {    while (n > 0) {        auto r = n % 10;        if (r == 0 || indexOf(digits, r) >= 0) {            return false;        }        le--;        digits[le] = r;        n /= 10;    }    return true;} int removeDigit(int[] digits, int le, int idx) {    enum pows = [ 1, 10, 100, 1_000, 10_000 ];     int sum = 0;    auto pow = pows[le - 2];    for (int i = 0; i < le; i++) {        if (i == idx) continue;        sum += digits[i] * pow;        pow /= 10;    }    return sum;} void main() {    auto lims = [ [ 12, 97 ], [ 123, 986 ], [ 1234, 9875 ], [ 12345, 98764 ] ];    int count;    int omitted;    for (int i = 0; i < lims.length; i++) {        auto nDigits = new int[i + 2];        auto dDigits = new int[i + 2];        for (int n = lims[i]; n <= lims[i]; n++) {            nDigits[] = 0;            bool nOk = getDigits(n, i + 2, nDigits);            if (!nOk) {                continue;            }            for (int d = n + 1; d <= lims[i] + 1; d++) {                dDigits[] = 0;                bool dOk = getDigits(d, i + 2, dDigits);                if (!dOk) {                    continue;                }                for (int nix = 0; nix < nDigits.length; nix++) {                    auto digit = nDigits[nix];                    auto dix = indexOf(dDigits, digit);                    if (dix >= 0) {                        auto rn = removeDigit(nDigits, i + 2, nix);                        auto rd = removeDigit(dDigits, i + 2, dix);                        if (cast(double)n / d == cast(double)rn / rd) {                            count[i]++;                            omitted[i][digit]++;                            if (count[i] <= 12) {                                writefln("%d/%d = %d/%d by omitting %d's", n, d, rn, rd, digit);                            }                        }                    }                }            }        }        writeln;    }     for (int i = 2; i <= 5; i++) {        writefln("There are %d %d-digit fractions of which:", count[i - 2], i);        for (int j = 1; j <= 9; j++) {            if (omitted[i - 2][j] == 0) {                continue;            }            writefln("%6s have %d's omitted", omitted[i - 2][j], j);        }        writeln;    }}`
Output:
```16/64 = 1/4 by omitting 6's
19/95 = 1/5 by omitting 9's
26/65 = 2/5 by omitting 6's
49/98 = 4/8 by omitting 9's

132/231 = 12/21 by omitting 3's
134/536 = 14/56 by omitting 3's
134/938 = 14/98 by omitting 3's
136/238 = 16/28 by omitting 3's
138/345 = 18/45 by omitting 3's
139/695 = 13/65 by omitting 9's
143/341 = 13/31 by omitting 4's
146/365 = 14/35 by omitting 6's
149/298 = 14/28 by omitting 9's
149/596 = 14/56 by omitting 9's
149/894 = 14/84 by omitting 9's
154/253 = 14/23 by omitting 5's

1234/4936 = 124/496 by omitting 3's
1239/6195 = 123/615 by omitting 9's
1246/3649 = 126/369 by omitting 4's
1249/2498 = 124/248 by omitting 9's
1259/6295 = 125/625 by omitting 9's
1279/6395 = 127/635 by omitting 9's
1283/5132 = 128/512 by omitting 3's
1297/2594 = 127/254 by omitting 9's
1297/3891 = 127/381 by omitting 9's
1298/2596 = 128/256 by omitting 9's
1298/3894 = 128/384 by omitting 9's
1298/5192 = 128/512 by omitting 9's

12349/24698 = 1234/2468 by omitting 9's
12356/67958 = 1236/6798 by omitting 5's
12358/14362 = 1258/1462 by omitting 3's
12358/15364 = 1258/1564 by omitting 3's
12358/17368 = 1258/1768 by omitting 3's
12358/19372 = 1258/1972 by omitting 3's
12358/21376 = 1258/2176 by omitting 3's
12358/25384 = 1258/2584 by omitting 3's
12359/61795 = 1235/6175 by omitting 9's
12364/32596 = 1364/3596 by omitting 2's
12379/61895 = 1237/6185 by omitting 9's
12386/32654 = 1386/3654 by omitting 2's

There are 4 2-digit fractions of which:
2 have 6's omitted
2 have 9's omitted

There are 122 3-digit fractions of which:
9 have 3's omitted
1 have 4's omitted
6 have 5's omitted
15 have 6's omitted
16 have 7's omitted
15 have 8's omitted
60 have 9's omitted

There are 660 4-digit fractions of which:
14 have 1's omitted
25 have 2's omitted
92 have 3's omitted
14 have 4's omitted
29 have 5's omitted
63 have 6's omitted
16 have 7's omitted
17 have 8's omitted
390 have 9's omitted

There are 5087 5-digit fractions of which:
75 have 1's omitted
40 have 2's omitted
376 have 3's omitted
78 have 4's omitted
209 have 5's omitted
379 have 6's omitted
591 have 7's omitted
351 have 8's omitted
2988 have 9's omitted```

## Go

### Version 1

This produces the stats for 5-digit fractions in less than 25 seconds but takes a much longer 15.5 minutes to process the 6-digit case. Timings are for an Intel Core i7-8565U machine.

`package main import (    "fmt"    "time") func indexOf(n int, s []int) int {    for i, j := range s {        if n == j {            return i        }    }    return -1} func getDigits(n, le int, digits []int) bool {    for n > 0 {        r := n % 10        if r == 0 || indexOf(r, digits) >= 0 {            return false        }        le--        digits[le] = r        n /= 10    }    return true} var pows = int{1, 10, 100, 1000, 10000} func removeDigit(digits []int, le, idx int) int {    sum := 0    pow := pows[le-2]    for i := 0; i < le; i++ {        if i == idx {            continue        }        sum += digits[i] * pow        pow /= 10    }    return sum} func main() {    start := time.Now()    lims := int{        {12, 97},        {123, 986},        {1234, 9875},        {12345, 98764},        {123456, 987653},    }    var count int    var omitted int    for i, lim := range lims {        nDigits := make([]int, i+2)        dDigits := make([]int, i+2)        blank := make([]int, i+2)        for n := lim; n <= lim; n++ {            copy(nDigits, blank)            nOk := getDigits(n, i+2, nDigits)            if !nOk {                continue            }            for d := n + 1; d <= lim+1; d++ {                copy(dDigits, blank)                dOk := getDigits(d, i+2, dDigits)                if !dOk {                    continue                }                for nix, digit := range nDigits {                    if dix := indexOf(digit, dDigits); dix >= 0 {                        rn := removeDigit(nDigits, i+2, nix)                        rd := removeDigit(dDigits, i+2, dix)                        if float64(n)/float64(d) == float64(rn)/float64(rd) {                            count[i]++                            omitted[i][digit]++                            if count[i] <= 12 {                                fmt.Printf("%d/%d = %d/%d by omitting %d's\n", n, d, rn, rd, digit)                            }                        }                    }                }            }        }        fmt.Println()    }     for i := 2; i <= 6; i++ {        fmt.Printf("There are %d %d-digit fractions of which:\n", count[i-2], i)        for j := 1; j <= 9; j++ {            if omitted[i-2][j] == 0 {                continue            }            fmt.Printf("%6d have %d's omitted\n", omitted[i-2][j], j)        }        fmt.Println()    }    fmt.Printf("Took %s\n", time.Since(start))}`
Output:
```16/64 = 1/4 by omitting 6's
19/95 = 1/5 by omitting 9's
26/65 = 2/5 by omitting 6's
49/98 = 4/8 by omitting 9's

132/231 = 12/21 by omitting 3's
134/536 = 14/56 by omitting 3's
134/938 = 14/98 by omitting 3's
136/238 = 16/28 by omitting 3's
138/345 = 18/45 by omitting 3's
139/695 = 13/65 by omitting 9's
143/341 = 13/31 by omitting 4's
146/365 = 14/35 by omitting 6's
149/298 = 14/28 by omitting 9's
149/596 = 14/56 by omitting 9's
149/894 = 14/84 by omitting 9's
154/253 = 14/23 by omitting 5's

1234/4936 = 124/496 by omitting 3's
1239/6195 = 123/615 by omitting 9's
1246/3649 = 126/369 by omitting 4's
1249/2498 = 124/248 by omitting 9's
1259/6295 = 125/625 by omitting 9's
1279/6395 = 127/635 by omitting 9's
1283/5132 = 128/512 by omitting 3's
1297/2594 = 127/254 by omitting 9's
1297/3891 = 127/381 by omitting 9's
1298/2596 = 128/256 by omitting 9's
1298/3894 = 128/384 by omitting 9's
1298/5192 = 128/512 by omitting 9's

12349/24698 = 1234/2468 by omitting 9's
12356/67958 = 1236/6798 by omitting 5's
12358/14362 = 1258/1462 by omitting 3's
12358/15364 = 1258/1564 by omitting 3's
12358/17368 = 1258/1768 by omitting 3's
12358/19372 = 1258/1972 by omitting 3's
12358/21376 = 1258/2176 by omitting 3's
12358/25384 = 1258/2584 by omitting 3's
12359/61795 = 1235/6175 by omitting 9's
12364/32596 = 1364/3596 by omitting 2's
12379/61895 = 1237/6185 by omitting 9's
12386/32654 = 1386/3654 by omitting 2's

123459/617295 = 12345/61725 by omitting 9's
123468/493872 = 12468/49872 by omitting 3's
123469/173524 = 12469/17524 by omitting 3's
123469/193546 = 12469/19546 by omitting 3's
123469/213568 = 12469/21568 by omitting 3's
123469/283645 = 12469/28645 by omitting 3's
123469/493876 = 12469/49876 by omitting 3's
123469/573964 = 12469/57964 by omitting 3's
123479/617395 = 12347/61735 by omitting 9's
123495/172893 = 12345/17283 by omitting 9's
123548/679514 = 12348/67914 by omitting 5's
123574/325786 = 13574/35786 by omitting 2's

There are 4 2-digit fractions of which:
2 have 6's omitted
2 have 9's omitted

There are 122 3-digit fractions of which:
9 have 3's omitted
1 have 4's omitted
6 have 5's omitted
15 have 6's omitted
16 have 7's omitted
15 have 8's omitted
60 have 9's omitted

There are 660 4-digit fractions of which:
14 have 1's omitted
25 have 2's omitted
92 have 3's omitted
14 have 4's omitted
29 have 5's omitted
63 have 6's omitted
16 have 7's omitted
17 have 8's omitted
390 have 9's omitted

There are 5087 5-digit fractions of which:
75 have 1's omitted
40 have 2's omitted
376 have 3's omitted
78 have 4's omitted
209 have 5's omitted
379 have 6's omitted
591 have 7's omitted
351 have 8's omitted
2988 have 9's omitted

There are 9778 6-digit fractions of which:
230 have 1's omitted
256 have 2's omitted
921 have 3's omitted
186 have 4's omitted
317 have 5's omitted
751 have 6's omitted
262 have 7's omitted
205 have 8's omitted
6650 have 9's omitted

Took 15m38.231915709s
```

### Version 2

Translation of: Phix

Rather than iterate through all numbers in the n-digit range and check if they contain unique non-zero digits, this generates all such numbers to start with which turns out to be a much more efficient approach - more than 20 times faster than before.

`package main import (    "fmt"    "time") type result struct {    n    int    nine int} func indexOf(n int, s []int) int {    for i, j := range s {        if n == j {            return i        }    }    return -1} func bIndexOf(b bool, s []bool) int {    for i, j := range s {        if b == j {            return i        }    }    return -1} func toNumber(digits []int, removeDigit int) int {    digits2 := digits    if removeDigit != 0 {        digits2 = make([]int, len(digits))        copy(digits2, digits)        d := indexOf(removeDigit, digits2)        copy(digits2[d:], digits2[d+1:])        digits2[len(digits2)-1] = 0        digits2 = digits2[:len(digits2)-1]    }    res := digits2    for i := 1; i < len(digits2); i++ {        res = res*10 + digits2[i]    }    return res} func nDigits(n int) []result {    var res []result    digits := make([]int, n)    var used bool    for i := 0; i < n; i++ {        digits[i] = i + 1        used[i] = true    }    for {        var nine int        for i := 0; i < len(used); i++ {            if used[i] {                nine[i] = toNumber(digits, i+1)            }        }        res = append(res, result{toNumber(digits, 0), nine})        found := false        for i := n - 1; i >= 0; i-- {            d := digits[i]            if !used[d-1] {                panic("something went wrong with 'used' array")            }            used[d-1] = false            for j := d; j < 9; j++ {                if !used[j] {                    used[j] = true                    digits[i] = j + 1                    for k := i + 1; k < n; k++ {                        digits[k] = bIndexOf(false, used[:]) + 1                        used[digits[k]-1] = true                    }                    found = true                    break                }            }            if found {                break            }        }        if !found {            break        }    }    return res} func main() {    start := time.Now()    for n := 2; n <= 5; n++ {        rs := nDigits(n)        count := 0        var omitted int        for i := 0; i < len(rs)-1; i++ {            xn, rn := rs[i].n, rs[i].nine            for j := i + 1; j < len(rs); j++ {                xd, rd := rs[j].n, rs[j].nine                for k := 0; k < 9; k++ {                    yn, yd := rn[k], rd[k]                    if yn != 0 && yd != 0 &&                        float64(xn)/float64(xd) == float64(yn)/float64(yd) {                        count++                        omitted[k]++                        if count <= 12 {                            fmt.Printf("%d/%d => %d/%d (removed %d)\n", xn, xd, yn, yd, k+1)                        }                    }                }            }        }        fmt.Printf("%d-digit fractions found:%d, omitted %v\n\n", n, count, omitted)    }    fmt.Printf("Took %s\n", time.Since(start))}`
Output:
```16/64 => 1/4 (removed 6)
19/95 => 1/5 (removed 9)
26/65 => 2/5 (removed 6)
49/98 => 4/8 (removed 9)
2-digit fractions found:4, omitted [0 0 0 0 0 2 0 0 2]

132/231 => 12/21 (removed 3)
134/536 => 14/56 (removed 3)
134/938 => 14/98 (removed 3)
136/238 => 16/28 (removed 3)
138/345 => 18/45 (removed 3)
139/695 => 13/65 (removed 9)
143/341 => 13/31 (removed 4)
146/365 => 14/35 (removed 6)
149/298 => 14/28 (removed 9)
149/596 => 14/56 (removed 9)
149/894 => 14/84 (removed 9)
154/253 => 14/23 (removed 5)
3-digit fractions found:122, omitted [0 0 9 1 6 15 16 15 60]

1234/4936 => 124/496 (removed 3)
1239/6195 => 123/615 (removed 9)
1246/3649 => 126/369 (removed 4)
1249/2498 => 124/248 (removed 9)
1259/6295 => 125/625 (removed 9)
1279/6395 => 127/635 (removed 9)
1283/5132 => 128/512 (removed 3)
1297/2594 => 127/254 (removed 9)
1297/3891 => 127/381 (removed 9)
1298/2596 => 128/256 (removed 9)
1298/3894 => 128/384 (removed 9)
1298/5192 => 128/512 (removed 9)
4-digit fractions found:660, omitted [14 25 92 14 29 63 16 17 390]

12349/24698 => 1234/2468 (removed 9)
12356/67958 => 1236/6798 (removed 5)
12358/14362 => 1258/1462 (removed 3)
12358/15364 => 1258/1564 (removed 3)
12358/17368 => 1258/1768 (removed 3)
12358/19372 => 1258/1972 (removed 3)
12358/21376 => 1258/2176 (removed 3)
12358/25384 => 1258/2584 (removed 3)
12359/61795 => 1235/6175 (removed 9)
12364/32596 => 1364/3596 (removed 2)
12379/61895 => 1237/6185 (removed 9)
12386/32654 => 1386/3654 (removed 2)
5-digit fractions found:5087, omitted [75 40 376 78 209 379 591 351 2988]

123459/617295 => 12345/61725 (removed 9)
123468/493872 => 12468/49872 (removed 3)
123469/173524 => 12469/17524 (removed 3)
123469/193546 => 12469/19546 (removed 3)
123469/213568 => 12469/21568 (removed 3)
123469/283645 => 12469/28645 (removed 3)
123469/493876 => 12469/49876 (removed 3)
123469/573964 => 12469/57964 (removed 3)
123479/617395 => 12347/61735 (removed 9)
123495/172893 => 12345/17283 (removed 9)
123548/679514 => 12348/67914 (removed 5)
123574/325786 => 13574/35786 (removed 2)
6-digit fractions found:9778, omitted [230 256 921 186 317 751 262 205 6650]

Took 42.251172302s
```

## Java

` import java.util.ArrayList;import java.util.Collections;import java.util.HashMap;import java.util.List;import java.util.Map; public class FractionReduction {     public static void main(String[] args) {        for ( int size = 2 ; size <= 5 ; size++ ) {            reduce(size);        }    }     private static void reduce(int numDigits) {        System.out.printf("Fractions with digits of length %d where cancellation is valid.  Examples:%n", numDigits);         //  Generate allowed numerator's and denominator's        int min = (int) Math.pow(10, numDigits-1);        int max = (int) Math.pow(10, numDigits) - 1;        List<Integer> values = new ArrayList<>();        for ( int number = min ; number <= max ; number++ ) {            if ( isValid(number) ) {                values.add(number);            }        }         Map<Integer,Integer> cancelCount = new HashMap<>();         int size = values.size();        int solutions = 0;        for ( int nIndex = 0 ; nIndex < size - 1 ; nIndex++ ) {            int numerator = values.get(nIndex);            //  Must be proper fraction            for ( int dIndex = nIndex + 1 ; dIndex < size ; dIndex++ ) {                int denominator = values.get(dIndex);                for ( int commonDigit : digitsInCommon(numerator, denominator) ) {                    int numRemoved = removeDigit(numerator, commonDigit);                    int denRemoved = removeDigit(denominator, commonDigit);                    if ( numerator * denRemoved == denominator * numRemoved ) {                        solutions++;                        cancelCount.merge(commonDigit, 1, (v1, v2) -> v1 + v2);                        if ( solutions <= 12 ) {                            System.out.printf("    When %d is removed, %d/%d = %d/%d%n", commonDigit, numerator, denominator, numRemoved, denRemoved);                        }                    }                }            }        }        System.out.printf("Number of fractions where cancellation is valid = %d.%n", solutions);        List<Integer> sorted = new ArrayList<>(cancelCount.keySet());        Collections.sort(sorted);        for ( int removed : sorted ) {            System.out.printf("    The digit %d was removed %d times.%n", removed, cancelCount.get(removed));        }        System.out.println();    }     private static int[] powers = new int[] {1, 10, 100, 1000, 10000, 100000};     //  Remove the specified digit.    private static int removeDigit(int n, int removed) {        int m = 0;        int pow = 0;        while ( n > 0 ) {            int r = n % 10;            if ( r != removed ) {                m = m + r*powers[pow];                pow++;            }            n /= 10;        }        return m;    }     //  Assumes no duplicate digits individually in n1 or n2 - part of task    private static List<Integer> digitsInCommon(int n1, int n2) {        int[] count = new int;        List<Integer> common = new ArrayList<>();        while ( n1 > 0 ) {            int r = n1 % 10;            count[r] += 1;            n1 /= 10;        }        while ( n2 > 0 ) {            int r = n2 % 10;            if ( count[r] > 0 ) {                common.add(r);            }            n2 /= 10;        }        return common;    }     //  No repeating digits, no digit is zero.    private static boolean isValid(int num) {        int[] count = new int;        while ( num > 0 ) {            int r = num % 10;            if ( r == 0 || count[r] == 1 ) {                return false;            }            count[r] = 1;            num /= 10;        }        return true;    } } `
Output:
```Fractions with digits of length 2 where cancellation is valid.  Examples:
When 6 is removed, 16/64 = 1/4
When 9 is removed, 19/95 = 1/5
When 6 is removed, 26/65 = 2/5
When 9 is removed, 49/98 = 4/8
Number of fractions where cancellation is valid = 4.
The digit 6 was removed 2 times.
The digit 9 was removed 2 times.

Fractions with digits of length 3 where cancellation is valid.  Examples:
When 3 is removed, 132/231 = 12/21
When 3 is removed, 134/536 = 14/56
When 3 is removed, 134/938 = 14/98
When 3 is removed, 136/238 = 16/28
When 3 is removed, 138/345 = 18/45
When 9 is removed, 139/695 = 13/65
When 4 is removed, 143/341 = 13/31
When 6 is removed, 146/365 = 14/35
When 9 is removed, 149/298 = 14/28
When 9 is removed, 149/596 = 14/56
When 9 is removed, 149/894 = 14/84
When 5 is removed, 154/253 = 14/23
Number of fractions where cancellation is valid = 122.
The digit 3 was removed 9 times.
The digit 4 was removed 1 times.
The digit 5 was removed 6 times.
The digit 6 was removed 15 times.
The digit 7 was removed 16 times.
The digit 8 was removed 15 times.
The digit 9 was removed 60 times.

Fractions with digits of length 4 where cancellation is valid.  Examples:
When 3 is removed, 1234/4936 = 124/496
When 9 is removed, 1239/6195 = 123/615
When 4 is removed, 1246/3649 = 126/369
When 9 is removed, 1249/2498 = 124/248
When 9 is removed, 1259/6295 = 125/625
When 9 is removed, 1279/6395 = 127/635
When 3 is removed, 1283/5132 = 128/512
When 9 is removed, 1297/2594 = 127/254
When 9 is removed, 1297/3891 = 127/381
When 9 is removed, 1298/2596 = 128/256
When 9 is removed, 1298/3894 = 128/384
When 9 is removed, 1298/5192 = 128/512
Number of fractions where cancellation is valid = 660.
The digit 1 was removed 14 times.
The digit 2 was removed 25 times.
The digit 3 was removed 92 times.
The digit 4 was removed 14 times.
The digit 5 was removed 29 times.
The digit 6 was removed 63 times.
The digit 7 was removed 16 times.
The digit 8 was removed 17 times.
The digit 9 was removed 390 times.

Fractions with digits of length 5 where cancellation is valid.  Examples:
When 9 is removed, 12349/24698 = 1234/2468
When 5 is removed, 12356/67958 = 1236/6798
When 3 is removed, 12358/14362 = 1258/1462
When 3 is removed, 12358/15364 = 1258/1564
When 3 is removed, 12358/17368 = 1258/1768
When 3 is removed, 12358/19372 = 1258/1972
When 3 is removed, 12358/21376 = 1258/2176
When 3 is removed, 12358/25384 = 1258/2584
When 9 is removed, 12359/61795 = 1235/6175
When 2 is removed, 12364/32596 = 1364/3596
When 9 is removed, 12379/61895 = 1237/6185
When 2 is removed, 12386/32654 = 1386/3654
Number of fractions where cancellation is valid = 5087.
The digit 1 was removed 75 times.
The digit 2 was removed 40 times.
The digit 3 was removed 376 times.
The digit 4 was removed 78 times.
The digit 5 was removed 209 times.
The digit 6 was removed 379 times.
The digit 7 was removed 591 times.
The digit 8 was removed 351 times.
The digit 9 was removed 2988 times.

```

## Julia

`using Combinatorics toi(set) = parse(Int, join(set, ""))drop1(c, set) = toi(filter(x -> x != c, set)) function anomalouscancellingfractions(numdigits)    ret = Vector{Tuple{Int, Int, Int, Int, Int}}()    for nset in permutations(1:9, numdigits), dset in permutations(1:9, numdigits)        if nset < dset # only proper fractions            for c in nset                if c in dset # a common digit exists                    n, d, nn, dd = toi(nset), toi(dset), drop1(c, nset), drop1(c, dset)                    if n // d == nn // dd # anomalous cancellation                        push!(ret, (n, d, nn, dd, c))                    end                end            end        end    end    retend function testfractionreduction(maxdigits=5)    for i in 2:maxdigits        results = anomalouscancellingfractions(i)        println("\nFor \$i digits, there were ", length(results),            " fractions with anomalous cancellation.")        numcounts = zeros(Int, 9)        for r in results            numcounts[r] += 1        end        for (j, count) in enumerate(numcounts)            count > 0 && println("The digit \$j was crossed out \$count times.")        end        println("Examples:")        for j in 1:min(length(results), 12)            r = results[j]            println(r, "/", r, " = ", r, "/", r, "   (\$(r) crossed out)")        end    endend testfractionreduction() `
Output:
```For 2 digits, there were 4 fractions with anomalous cancellation.
The digit 6 was crossed out 2 times.
The digit 9 was crossed out 2 times.
Examples:
16/64 = 1/4   (6 crossed out)
19/95 = 1/5   (9 crossed out)
26/65 = 2/5   (6 crossed out)
49/98 = 4/8   (9 crossed out)

For 3 digits, there were 122 fractions with anomalous cancellation.
The digit 3 was crossed out 9 times.
The digit 4 was crossed out 1 times.
The digit 5 was crossed out 6 times.
The digit 6 was crossed out 15 times.
The digit 7 was crossed out 16 times.
The digit 8 was crossed out 15 times.
The digit 9 was crossed out 60 times.
Examples:
132/231 = 12/21   (3 crossed out)
134/536 = 14/56   (3 crossed out)
134/938 = 14/98   (3 crossed out)
136/238 = 16/28   (3 crossed out)
138/345 = 18/45   (3 crossed out)
139/695 = 13/65   (9 crossed out)
143/341 = 13/31   (4 crossed out)
146/365 = 14/35   (6 crossed out)
149/298 = 14/28   (9 crossed out)
149/596 = 14/56   (9 crossed out)
149/894 = 14/84   (9 crossed out)
154/253 = 14/23   (5 crossed out)

For 4 digits, there were 660 fractions with anomalous cancellation.
The digit 1 was crossed out 14 times.
The digit 2 was crossed out 25 times.
The digit 3 was crossed out 92 times.
The digit 4 was crossed out 14 times.
The digit 5 was crossed out 29 times.
The digit 6 was crossed out 63 times.
The digit 7 was crossed out 16 times.
The digit 8 was crossed out 17 times.
The digit 9 was crossed out 390 times.
Examples:
1234/4936 = 124/496   (3 crossed out)
1239/6195 = 123/615   (9 crossed out)
1246/3649 = 126/369   (4 crossed out)
1249/2498 = 124/248   (9 crossed out)
1259/6295 = 125/625   (9 crossed out)
1279/6395 = 127/635   (9 crossed out)
1283/5132 = 128/512   (3 crossed out)
1297/2594 = 127/254   (9 crossed out)
1297/3891 = 127/381   (9 crossed out)
1298/2596 = 128/256   (9 crossed out)
1298/3894 = 128/384   (9 crossed out)
1298/5192 = 128/512   (9 crossed out)

For 5 digits, there were 5087 fractions with anomalous cancellation.
The digit 1 was crossed out 75 times.
The digit 2 was crossed out 40 times.
The digit 3 was crossed out 376 times.
The digit 4 was crossed out 78 times.
The digit 5 was crossed out 209 times.
The digit 6 was crossed out 379 times.
The digit 7 was crossed out 591 times.
The digit 8 was crossed out 351 times.
The digit 9 was crossed out 2988 times.
Examples:
12349/24698 = 1234/2468   (9 crossed out)
12356/67958 = 1236/6798   (5 crossed out)
12358/14362 = 1258/1462   (3 crossed out)
12358/15364 = 1258/1564   (3 crossed out)
12358/17368 = 1258/1768   (3 crossed out)
12358/19372 = 1258/1972   (3 crossed out)
12358/21376 = 1258/2176   (3 crossed out)
12358/25384 = 1258/2584   (3 crossed out)
12359/61795 = 1235/6175   (9 crossed out)
12364/32596 = 1364/3596   (2 crossed out)
12379/61895 = 1237/6185   (9 crossed out)
12386/32654 = 1386/3654   (2 crossed out)
```

## Kotlin

Translation of: Go
`fun indexOf(n: Int, s: IntArray): Int {    for (i_j in s.withIndex()) {        if (n == i_j.value) {            return i_j.index        }    }    return -1} fun getDigits(n: Int, le: Int, digits: IntArray): Boolean {    var mn = n    var mle = le    while (mn > 0) {        val r = mn % 10        if (r == 0 || indexOf(r, digits) >= 0) {            return false        }        mle--        digits[mle] = r        mn /= 10    }    return true} val pows = intArrayOf(1, 10, 100, 1_000, 10_000) fun removeDigit(digits: IntArray, le: Int, idx: Int): Int {    var sum = 0    var pow = pows[le - 2]    for (i in 0 until le) {        if (i == idx) {            continue        }        sum += digits[i] * pow        pow /= 10    }    return sum} fun main() {    val lims = listOf(        Pair(12, 97),        Pair(123, 986),        Pair(1234, 9875),        Pair(12345, 98764)    )    val count = IntArray(5)    var omitted = arrayOf<Array<Int>>()    for (i in 0 until 5) {        var array = arrayOf<Int>()        for (j in 0 until 10) {            array += 0        }        omitted += array    }    for (i_lim in lims.withIndex()) {        val i = i_lim.index        val lim = i_lim.value         val nDigits = IntArray(i + 2)        val dDigits = IntArray(i + 2)        val blank = IntArray(i + 2) { 0 }        for (n in lim.first..lim.second) {            blank.copyInto(nDigits)            val nOk = getDigits(n, i + 2, nDigits)            if (!nOk) {                continue            }            for (d in n + 1..lim.second + 1) {                blank.copyInto(dDigits)                val dOk = getDigits(d, i + 2, dDigits)                if (!dOk) {                    continue                }                for (nix_digit in nDigits.withIndex()) {                    val dix = indexOf(nix_digit.value, dDigits)                    if (dix >= 0) {                        val rn = removeDigit(nDigits, i + 2, nix_digit.index)                        val rd = removeDigit(dDigits, i + 2, dix)                        if (n.toDouble() / d.toDouble() == rn.toDouble() / rd.toDouble()) {                            count[i]++                            omitted[i][nix_digit.value]++                            if (count[i] <= 12) {                                println("\$n/\$d = \$rn/\$rd by omitting \${nix_digit.value}'s")                            }                        }                    }                }            }        }        println()    }     for (i in 2..5) {        println("There are \${count[i - 2]} \$i-digit fractions of which:")        for (j in 1..9) {            if (omitted[i - 2][j] == 0) {                continue            }            println("%6d have %d's omitted".format(omitted[i - 2][j], j))        }        println()    }}`
Output:
```16/64 = 1/4 by omitting 6's
19/95 = 1/5 by omitting 9's
26/65 = 2/5 by omitting 6's
49/98 = 4/8 by omitting 9's

132/231 = 12/21 by omitting 3's
134/536 = 14/56 by omitting 3's
134/938 = 14/98 by omitting 3's
136/238 = 16/28 by omitting 3's
138/345 = 18/45 by omitting 3's
139/695 = 13/65 by omitting 9's
143/341 = 13/31 by omitting 4's
146/365 = 14/35 by omitting 6's
149/298 = 14/28 by omitting 9's
149/596 = 14/56 by omitting 9's
149/894 = 14/84 by omitting 9's
154/253 = 14/23 by omitting 5's

1234/4936 = 124/496 by omitting 3's
1239/6195 = 123/615 by omitting 9's
1246/3649 = 126/369 by omitting 4's
1249/2498 = 124/248 by omitting 9's
1259/6295 = 125/625 by omitting 9's
1279/6395 = 127/635 by omitting 9's
1283/5132 = 128/512 by omitting 3's
1297/2594 = 127/254 by omitting 9's
1297/3891 = 127/381 by omitting 9's
1298/2596 = 128/256 by omitting 9's
1298/3894 = 128/384 by omitting 9's
1298/5192 = 128/512 by omitting 9's

12349/24698 = 1234/2468 by omitting 9's
12356/67958 = 1236/6798 by omitting 5's
12358/14362 = 1258/1462 by omitting 3's
12358/15364 = 1258/1564 by omitting 3's
12358/17368 = 1258/1768 by omitting 3's
12358/19372 = 1258/1972 by omitting 3's
12358/21376 = 1258/2176 by omitting 3's
12358/25384 = 1258/2584 by omitting 3's
12359/61795 = 1235/6175 by omitting 9's
12364/32596 = 1364/3596 by omitting 2's
12379/61895 = 1237/6185 by omitting 9's
12386/32654 = 1386/3654 by omitting 2's

There are 4 2-digit fractions of which:
2 have 6's omitted
2 have 9's omitted

There are 122 3-digit fractions of which:
9 have 3's omitted
1 have 4's omitted
6 have 5's omitted
15 have 6's omitted
16 have 7's omitted
15 have 8's omitted
60 have 9's omitted

There are 660 4-digit fractions of which:
14 have 1's omitted
25 have 2's omitted
92 have 3's omitted
14 have 4's omitted
29 have 5's omitted
63 have 6's omitted
16 have 7's omitted
17 have 8's omitted
390 have 9's omitted

There are 5087 5-digit fractions of which:
75 have 1's omitted
40 have 2's omitted
376 have 3's omitted
78 have 4's omitted
209 have 5's omitted
379 have 6's omitted
591 have 7's omitted
351 have 8's omitted
2988 have 9's omitted```

## MiniZinc

### The Model

` %Fraction Reduction. Nigel Galloway, September 5th., 2019include "alldifferent.mzn"; include "member.mzn";int: S;array [1..9] of int: Pn=[1,10,100,1000,10000,100000,1000000,10000000,100000000];array [1..S] of var 1..9: Nz; constraint alldifferent(Nz);array [1..S] of var 1..9: Gz; constraint alldifferent(Gz);var  int: n; constraint n=sum(n in 1..S)(Nz[n]*Pn[n]);var  int: i; constraint i=sum(n in 1..S)(Gz[n]*Pn[n]); constraint n<i; constraint n*g=i*e;var  int: g; constraint g=sum(n in 1..S)(if n=a then 0 elseif n>a then Gz[n]*Pn[n-1] else Gz[n]*Pn[n] endif);var  int: e; constraint e=sum(n in 1..S)(if n=l then 0 elseif n>l then Nz[n]*Pn[n-1] else Nz[n]*Pn[n] endif);var 1..S: l; constraint Nz[l]=w;var 1..S: a; constraint Gz[a]=w;var 1..9: w; constraint member(Nz,w) /\ member(Gz,w); output [show(n)++"/"++show(i)++" becomes "++show(e)++"/"++show(g)++" when "++show(w)++" is omitted"] `

Displaying 12 solutions
minizinc --num-solutions 12 -DS=2
Output:
```16/64 becomes 1/4 when 6 is omitted
----------
26/65 becomes 2/5 when 6 is omitted
----------
19/95 becomes 1/5 when 9 is omitted
----------
49/98 becomes 4/8 when 9 is omitted
----------
==========
```
minizinc --num-solutions 12 -DS=3
Output:
```132/231 becomes 12/21 when 3 is omitted
----------
134/536 becomes 14/56 when 3 is omitted
----------
134/938 becomes 14/98 when 3 is omitted
----------
136/238 becomes 16/28 when 3 is omitted
----------
138/345 becomes 18/45 when 3 is omitted
----------
139/695 becomes 13/65 when 9 is omitted
----------
143/341 becomes 13/31 when 4 is omitted
----------
146/365 becomes 14/35 when 6 is omitted
----------
149/298 becomes 14/28 when 9 is omitted
----------
149/596 becomes 14/56 when 9 is omitted
----------
149/894 becomes 14/84 when 9 is omitted
----------
154/253 becomes 14/23 when 5 is omitted
----------
```
minizinc --num-solutions 12 -DS=4
Output:
```2147/3164 becomes 247/364 when 1 is omitted
----------
2314/3916 becomes 234/396 when 1 is omitted
----------
2147/5198 becomes 247/598 when 1 is omitted
----------
3164/5198 becomes 364/598 when 1 is omitted
----------
2314/6319 becomes 234/639 when 1 is omitted
----------
3916/6319 becomes 396/639 when 1 is omitted
----------
5129/7136 becomes 529/736 when 1 is omitted
----------
3129/7152 becomes 329/752 when 1 is omitted
----------
4913/7514 becomes 493/754 when 1 is omitted
----------
7168/8176 becomes 768/876 when 1 is omitted
----------
5129/9143 becomes 529/943 when 1 is omitted
----------
7136/9143 becomes 736/943 when 1 is omitted
----------
```
minizinc --num-solutions 12 -DS=5
Output:
```21356/31472 becomes 2356/3472 when 1 is omitted
----------
21394/31528 becomes 2394/3528 when 1 is omitted
----------
21546/31752 becomes 2546/3752 when 1 is omitted
----------
21679/31948 becomes 2679/3948 when 1 is omitted
----------
21698/31976 becomes 2698/3976 when 1 is omitted
----------
25714/34615 becomes 2574/3465 when 1 is omitted
----------
27615/34716 becomes 2765/3476 when 1 is omitted
----------
25917/34719 becomes 2597/3479 when 1 is omitted
----------
25916/36518 becomes 2596/3658 when 1 is omitted
----------
31276/41329 becomes 3276/4329 when 1 is omitted
----------
21375/41625 becomes 2375/4625 when 1 is omitted
----------
31584/41736 becomes 3584/4736 when 1 is omitted
----------
```
minizinc --num-solutions 12 -DS=6
Output:
```123495/172893 becomes 12345/17283 when 9 is omitted
----------
123594/164792 becomes 12354/16472 when 9 is omitted
----------
123654/163758 becomes 12654/16758 when 3 is omitted
----------
124678/135679 becomes 12478/13579 when 6 is omitted
----------
124768/164872 becomes 12768/16872 when 4 is omitted
----------
125349/149352 becomes 12549/14952 when 3 is omitted
----------
125394/146293 becomes 12534/14623 when 9 is omitted
----------
125937/127936 becomes 12537/12736 when 9 is omitted
----------
125694/167592 becomes 12564/16752 when 9 is omitted
----------
125769/135786 becomes 12769/13786 when 5 is omitted
----------
125769/165837 becomes 12769/16837 when 5 is omitted
----------
125934/146923 becomes 12534/14623 when 9 is omitted
----------
```
Count number of solutions
minizinc --all-solutions -s -DS=3
Output:
```%%%mzn-stat: nSolutions=122
```
minizinc --all-solutions -s -DS=4
Output:
```%%%mzn-stat: nSolutions=660
```
minizinc --all-solutions -s -DS=5
Output:
```%%%mzn-stat: nSolutions=5087
```

## Pascal

Works with: Free Pascal

Using a permutation k out of n with k <= n
Inserting a record with this number and all numbers with one digit removed of that number.So only once calculated.Trade off is big size and no cache friendly local access.

` program FracRedu;{\$IFDEF FPC}  {\$MODE DELPHI}  {\$OPTIMIZATION ON,ALL}{\$ELSE}  {\$APPTYPE CONSOLE}{\$ENDIF}uses  SysUtils; type  tdigit = 0..9;const  cMaskDgt: array [tdigit] of Uint32 = (1, 2, 4, 8, 16, 32, 64, 128, 256, 512    {,1024,2048,4096,8193,16384,32768});  cMaxDigits = High(tdigit);type  tPermfield = array[tdigit] of uint32;  tpPermfield = ^tPermfield;   tDigitCnt = array[tdigit] of Uint32;   tErg = record           numUsedDigits : Uint32;           numUnusedDigit : array[tdigit] of Uint32;           numNormal : Uint64;// so sqr of number stays in Uint64           dummy : array[0..7] of byte;//-> sizeof(tErg) = 64         end;  tpErg = ^tErg;var  Erg: array of tErg;  pf_x, pf_y: tPermfield;  DigitCnt :tDigitCnt;  permcnt, UsedDigits,Anzahl: NativeUint;   function Fakultaet(i: integer): integer;  begin    Result := 1;    while i > 1 do    begin      Result := Result * i;      Dec(i);    end;  end;   procedure OutErg(dgt: Uint32;pi,pJ:tpErg);  begin    writeln(dgt:3,'  ', pi^.numUnusedDigit[dgt],'/',pj^.numUnusedDigit[dgt]            ,' = ',pi^.numNormal,'/',pj^.numNormal);  end;   function Check(pI,pJ : tpErg;Nud :Word):integer;  var    dgt: NativeInt;  Begin    result := 0;    dgt := 1;    NUD := NUD SHR 1;    repeat      IF NUD AND 1 <> 0 then      Begin        If  pI^.numNormal*pJ^.numUnusedDigit[dgt] = pJ^.numNormal*pI^.numUnusedDigit[dgt] then        Begin          inc(result);          inc(DigitCnt[dgt]);          IF Anzahl < 110 then            OutErg(dgt,pI,pJ);        end;      end;      inc(dgt);      NUD := NUD SHR 1;    until NUD = 0;  end;   procedure CheckWithOne(pI : tpErg;j,Nud:Uint32);  var    pJ : tpErg;    l : NativeUInt;  Begin    pJ := pI;    if UsedDigits <5 then    Begin      for j := j+1 to permcnt do      begin        inc(pJ);        //digits used by both numbers        l := NUD AND pJ^.numUsedDigits;        IF l <> 0 then          inc(Anzahl,Check(pI,pJ,l));      end;    end    else    Begin      for j := j+1 to permcnt do      begin        inc(pJ);        l := NUD AND pJ^.numUsedDigits;        inc(Anzahl,Check(pI,pJ,l));      end;    end;  end;   procedure SearchMultiple;  var    pI : tpErg;    i : NativeUInt;  begin    pI := @Erg;    for i := 0 to permcnt do    Begin      CheckWithOne(pI,i,pI^.numUsedDigits);      inc(pI);    end;  end;   function BinomCoeff(n, k: byte): longint;  var    i: longint;  begin    {n ueber k  = n ueber (n-k) , also kuerzere Version waehlen}    if k > n div 2 then      k := n - k;    Result := 1;    if k <= n then      for i := 1 to k do        Result := Result * (n - i + 1) div i;{geht immer  ohne Rest }  end;   procedure InsertToErg(var E: tErg; const x: tPermfield);  var    n : Uint64;    k,i,j,dgt,nud: NativeInt;  begin    // k of PermKoutofN is reduced by one for 9 digits    k := UsedDigits;    n := 0;    nud := 0;    for i := 1 to k do    begin      dgt := x[i];      nud := nud or cMaskDgt[dgt];      n := n * 10 + dgt;    end;    with E do    begin      numUsedDigits := nud;      numNormal := n;    end;    //calc all numbers with one removed digit    For J := k downto 1 do    Begin      n := 0;      for i := 1 to j-1 do        n := n * 10 + x[i];      for i := j+1 to k do        n := n * 10 + x[i];      E.numUnusedDigit[x[j]] := n;    end;  end;   procedure PermKoutofN(k, n: nativeInt);  var    x, y: tpPermfield;    i, yi, tmp: NativeInt;  begin    //initialise    x := @pf_x;    y := @pf_y;    permcnt := 0;    if k > n then      k := n;    if k = n then      k := k - 1;    for i := 1 to n do      x^[i] := i;    for i := 1 to k do      y^[i] := i;     InserttoErg(Erg[permcnt], x^);    i := k;    repeat      yi := y^[i];      if yi < n then      begin        Inc(permcnt);        Inc(yi);        y^[i] := yi;        tmp := x^[i];        x^[i] := x^[yi];        x^[yi] := tmp;        i := k;        InserttoErg(Erg[permcnt], x^);      end      else      begin        repeat          tmp := x^[i];          x^[i] := x^[yi];          x^[yi] := tmp;          Dec(yi);        until yi <= i;        y^[i] := yi;        Dec(i);      end;    until (i = 0);  end;   procedure OutDigitCount;   var     i : tDigit;  Begin    writeln('omitted digits 1 to 9');    For i := 1 to 9do      write(DigitCnt[i]:UsedDigits);    writeln;  end;   procedure ClearDigitCount;   var     i : tDigit;  Begin    For i := low(DigitCnt) to high(DigitCnt) do      DigitCnt[i] := 0;  end; var  t1, t0: TDateTime;begin  For UsedDigits := 8 to 9 do  Begin    writeln('Used digits ',UsedDigits);    T0 := now;    ClearDigitCount;    setlength(Erg, Fakultaet(UsedDigits) * BinomCoeff(cMaxDigits, UsedDigits));    Anzahl := 0;    permcnt := 0;    PermKoutOfN(UsedDigits, cMaxDigits);    SearchMultiple;    T1 := now;    writeln('Found solutions ',Anzahl);    OutDigitCount;    writeln('time taken ',FormatDateTime('HH:NN:SS.zzz', T1 - T0));    setlength(Erg, 0);    writeln;  end;end.`
Output:
```{ /* inserted by hand                            / solutions
Used digits 2 count of different numbers 72      /     4
Used digits 3 count of different numbers 504     /   122
Used digits 4 count of different numbers 3024    /   660
Used digits 5 count of different numbers 15120   /  5087
Used digits 6 count of different numbers 60480   /  9778
Used digits 7 count of different numbers 181440  / 40163
Used digits 8 count of different numbers 362880  / 17722
Used digits 9 count of different numbers 362880  / 92413
*/ }

Used digits 2
6  1/4 = 16/64
9  1/5 = 19/95
6  2/5 = 26/65
9  4/8 = 49/98
Found solutions 4
omitted digits 1 to 9
0 0 0 0 0 2 0 0 2
time taken 00:00:00.000

Used digits 3
3  12/21 = 132/231
3  14/56 = 134/536
3  14/98 = 134/938
3  16/28 = 136/238
3  18/45 = 138/345
9  13/65 = 139/695
4  13/31 = 143/341
6  14/35 = 146/365
9  14/28 = 149/298
9  14/56 = 149/596
9  14/84 = 149/894
5  14/23 = 154/253
Found solutions 122
omitted digits 1 to 9
0  0  9  1  6 15 16 15 60
time taken 00:00:00.004

Used digits 4
3  124/496 = 1234/4936
9  123/615 = 1239/6195
4  126/369 = 1246/3649
9  124/248 = 1249/2498
9  125/625 = 1259/6295
9  127/635 = 1279/6395
3  128/512 = 1283/5132
9  127/254 = 1297/2594
9  127/381 = 1297/3891
9  128/256 = 1298/2596
9  128/384 = 1298/3894
9  128/512 = 1298/5192
Found solutions 660
omitted digits 1 to 9
14  25  92  14  29  63  16  17 390
time taken 00:00:00.060

Used digits 5
9  1234/2468 = 12349/24698
5  1236/6798 = 12356/67958
3  1258/1462 = 12358/14362
3  1258/1564 = 12358/15364
3  1258/1768 = 12358/17368
3  1258/1972 = 12358/19372
3  1258/2176 = 12358/21376
3  1258/2584 = 12358/25384
9  1235/6175 = 12359/61795
2  1364/3596 = 12364/32596
9  1237/6185 = 12379/61895
2  1386/3654 = 12386/32654
Found solutions 5087
omitted digits 1 to 9
75   40  376   78  209  379  591  351 2988
time taken 00:00:01.787

Used digits 6
9  12345/61725 = 123459/617295
3  12468/49872 = 123468/493872
3  12469/17524 = 123469/173524
3  12469/19546 = 123469/193546
3  12469/21568 = 123469/213568
3  12469/28645 = 123469/283645
3  12469/49876 = 123469/493876
3  12469/57964 = 123469/573964
9  12347/61735 = 123479/617395
9  12345/17283 = 123495/172893
5  12348/67914 = 123548/679514
2  13574/35786 = 123574/325786
Found solutions 9778
omitted digits 1 to 9
230   256   921   186   317   751   262   205  6650
time taken 00:00:31.858

Used digits 7
3  124569/498276 = 1234569/4938276
3  124579/195286 = 1234579/1935286
3  124579/245791 = 1234579/2435791
3  124579/286195 = 1234579/2836195
3  124579/457912 = 1234579/4537912
3  124579/528619 = 1234579/5238619
3  124579/579124 = 1234579/5739124
3  124579/619528 = 1234579/6139528
9  123457/617285 = 1234579/6172895
9  123457/617285 = 1234597/6172985
9  123465/617325 = 1234659/6173295
3  124678/498712 = 1234678/4938712
Found solutions 40163
omitted digits 1 to 9
333    191   1368    278    498   1094   3657   1434  31310
time taken 00:04:54.703

Used digits 8
3  1245679/2457691 = 12345679/24357691
6  1234579/2435791 = 12345679/24357691
3  1245679/4982716 = 12345679/49382716
3  1245679/6194728 = 12345679/61394728
9  1234567/6172835 = 12345679/61728395
3  1245689/4982756 = 12345689/49382756
9  1234567/6172835 = 12345967/61729835
9  1234657/6173285 = 12346579/61732895
9  1234657/6173285 = 12346597/61732985
3  1246789/4987156 = 12346789/49387156
9  1234685/6173425 = 12346859/61734295
3  1246879/4987516 = 12346879/49387516
Found solutions 17233
omitted digits 1 to 9
247     233     888     288     355     710     425     193   13894
time taken 00:18:58.784

Used digits 9
3  12456789/49827156 = 123456789/493827156
3  12456879/49827516 = 123456879/493827516
9  12345687/61728435 = 123456879/617284395
9  12345687/61728435 = 123456987/617284935
9  12345687/61728435 = 123459687/617298435
9  12346857/61734285 = 123468579/617342895
9  12346857/61734285 = 123468597/617342985
9  12346857/61734285 = 123469857/617349285
9  12347685/61738425 = 123476859/617384295
9  12347685/61738425 = 123476985/617384925
5  12347896/67913428 = 123478956/679134258
9  12347685/61738425 = 123479685/617398425
Found solutions 92413
omitted digits 1 to 9
266      110     1008      131      324      737      300      159    89378
time taken 00:13:04.511

/*
go version go1.10.3 gccgo (Debian 8.3.0-6) 8.3.0 linux/amd64
6-digit fractions found:9778, omitted [230 256 921 186 317 751 262 205 6650]

Took 1m38.85577279s
*/```

## Perl

Translation of: Raku
`use strict;use warnings;use feature 'say';use List::Util qw<sum uniq uniqnum head tail>; for my \$exp (map { \$_ - 1 } <2 3 4>) {    my %reduced;    my \$start = sum map { 10 ** \$_ * (\$exp - \$_ + 1) } 0..\$exp;    my \$end   = 10**(\$exp+1) - -1 + sum map { 10 ** \$_ * (\$exp - \$_) } 0..\$exp-1;     for my \$den (\$start .. \$end-1) {        next if \$den =~ /0/ or (uniqnum split '', \$den) <= \$exp;        for my \$num (\$start .. \$den-1) {            next if \$num =~ /0/ or (uniqnum split '', \$num) <= \$exp;            my %i;            map { \$i{\$_}++ } (uniq head -1, split '',\$den), uniq tail -1, split '',\$num;            my @set = grep { \$_ if \$i{\$_} > 1 } keys %i;            next if @set < 1;            for (@set) {                (my \$ne = \$num) =~ s/\$_//;                (my \$de = \$den) =~ s/\$_//;                if (\$ne/\$de == \$num/\$den) {                    \$reduced{"\$num/\$den:\$_"} = "\$ne/\$de";                }            }        }    }    my \$digit = \$exp + 1;    say "\n" . +%reduced . " \$digit-digit reducible fractions:";    for my \$n (1..9) {        my \$cnt = scalar grep { /:\$n/ } keys %reduced;        say "\$cnt with removed \$n" if \$cnt;    }    say "\n  12 (or all, if less) \$digit-digit reducible fractions:";    for my \$f (head 12, sort keys %reduced) {        printf "    %s => %s removed %s\n", substr(\$f,0,\$digit*2+1), \$reduced{\$f}, substr(\$f,-1)    }}`
Output:
```4 2-digit reducible fractions:
2 with removed 6
2 with removed 9

12 (or all, if less) 2-digit reducible fractions:
16/64 => 1/4 removed 6
19/95 => 1/5 removed 9
26/65 => 2/5 removed 6
49/98 => 4/8 removed 9

122 3-digit reducible fractions:
9 with removed 3
1 with removed 4
6 with removed 5
15 with removed 6
16 with removed 7
15 with removed 8
60 with removed 9

12 (or all, if less) 3-digit reducible fractions:
132/231 => 12/21 removed 3
134/536 => 14/56 removed 3
134/938 => 14/98 removed 3
136/238 => 16/28 removed 3
138/345 => 18/45 removed 3
139/695 => 13/65 removed 9
143/341 => 13/31 removed 4
146/365 => 14/35 removed 6
149/298 => 14/28 removed 9
149/596 => 14/56 removed 9
149/894 => 14/84 removed 9
154/253 => 14/23 removed 5

660 4-digit reducible fractions:
14 with removed 1
25 with removed 2
92 with removed 3
14 with removed 4
29 with removed 5
63 with removed 6
16 with removed 7
17 with removed 8
390 with removed 9

12 (or all, if less) 4-digit reducible fractions:
1234/4936 => 124/496 removed 3
1239/6195 => 123/615 removed 9
1246/3649 => 126/369 removed 4
1249/2498 => 124/248 removed 9
1259/6295 => 125/625 removed 9
1279/6395 => 127/635 removed 9
1283/5132 => 128/512 removed 3
1297/2594 => 127/254 removed 9
1297/3891 => 127/381 removed 9
1298/2596 => 128/256 removed 9
1298/3894 => 128/384 removed 9
1298/5192 => 128/512 removed 9```

## Phix

`function to_n(sequence digits, integer remove_digit=0)    if remove_digit!=0 then        integer d = find(remove_digit,digits)        digits[d..d] = {}    end if    integer res = digits    for i=2 to length(digits) do        res = res*10+digits[i]    end for    return resend function function ndigits(integer n)-- generate numbers with unique digits efficiently -- and store them in an array for multiple re-use,-- along with an array of the removed-digit values.    sequence res = {},             digits = tagset(n),             used = repeat(1,n)&repeat(0,9-n)    while true do        sequence nine = repeat(0,9)        for i=1 to length(used) do            if used[i] then                nine[i] = to_n(digits,i)            end if        end for        res = append(res,{to_n(digits),nine})        bool found = false        for i=n to 1 by -1 do            integer d = digits[i]            if not used[d] then ?9/0 end if            used[d] = 0            for j=d+1 to 9 do                if not used[j] then                    used[j] = 1                    digits[i] = j                    for k=i+1 to n do                        digits[k] = find(0,used)                        used[digits[k]] = 1                    end for                    found = true                    exit                end if            end for            if found then exit end if        end for        if not found then exit end if    end while    return resend function atom t0 = time(),     t1 = time()+1for n=2 to 6 do    sequence d = ndigits(n)    integer count = 0    sequence omitted = repeat(0,9)    for i=1 to length(d)-1 do        {integer xn, sequence rn} = d[i]        for j=i+1 to length(d) do            {integer xd, sequence rd} = d[j]            for k=1 to 9 do                integer yn = rn[k], yd = rd[k]                if yn!=0 and yd!=0 and xn/xd = yn/yd then                    count += 1                    omitted[k] += 1                    if count<=12 then                        printf(1,"%d/%d => %d/%d (removed %d)\n",{xn,xd,yn,yd,k})                    elsif time()>t1 then                        printf(1,"working (%d/%d)...\r",{i,length(d)})                        t1 = time()+1                    end if                end if            end for        end for    end for    printf(1,"%d-digit fractions found:%d, omitted %v\n\n",{n,count,omitted})end for?elapsed(time()-t0)`
Output:
```16/64 => 1/4 (removed 6)
19/95 => 1/5 (removed 9)
26/65 => 2/5 (removed 6)
49/98 => 4/8 (removed 9)
2-digit fractions found:4, omitted {0,0,0,0,0,2,0,0,2}

132/231 => 12/21 (removed 3)
134/536 => 14/56 (removed 3)
134/938 => 14/98 (removed 3)
136/238 => 16/28 (removed 3)
138/345 => 18/45 (removed 3)
139/695 => 13/65 (removed 9)
143/341 => 13/31 (removed 4)
146/365 => 14/35 (removed 6)
149/298 => 14/28 (removed 9)
149/596 => 14/56 (removed 9)
149/894 => 14/84 (removed 9)
154/253 => 14/23 (removed 5)
3-digit fractions found:122, omitted {0,0,9,1,6,15,16,15,60}

1234/4936 => 124/496 (removed 3)
1239/6195 => 123/615 (removed 9)
1246/3649 => 126/369 (removed 4)
1249/2498 => 124/248 (removed 9)
1259/6295 => 125/625 (removed 9)
1279/6395 => 127/635 (removed 9)
1283/5132 => 128/512 (removed 3)
1297/2594 => 127/254 (removed 9)
1297/3891 => 127/381 (removed 9)
1298/2596 => 128/256 (removed 9)
1298/3894 => 128/384 (removed 9)
1298/5192 => 128/512 (removed 9)
4-digit fractions found:660, omitted {14,25,92,14,29,63,16,17,390}

12349/24698 => 1234/2468 (removed 9)
12356/67958 => 1236/6798 (removed 5)
12358/14362 => 1258/1462 (removed 3)
12358/15364 => 1258/1564 (removed 3)
12358/17368 => 1258/1768 (removed 3)
12358/19372 => 1258/1972 (removed 3)
12358/21376 => 1258/2176 (removed 3)
12358/25384 => 1258/2584 (removed 3)
12359/61795 => 1235/6175 (removed 9)
12364/32596 => 1364/3596 (removed 2)
12379/61895 => 1237/6185 (removed 9)
12386/32654 => 1386/3654 (removed 2)
5-digit fractions found:5087, omitted {75,40,376,78,209,379,591,351,2988}

123459/617295 => 12345/61725 (removed 9)
123468/493872 => 12468/49872 (removed 3)
123469/173524 => 12469/17524 (removed 3)
123469/193546 => 12469/19546 (removed 3)
123469/213568 => 12469/21568 (removed 3)
123469/283645 => 12469/28645 (removed 3)
123469/493876 => 12469/49876 (removed 3)
123469/573964 => 12469/57964 (removed 3)
123479/617395 => 12347/61735 (removed 9)
123495/172893 => 12345/17283 (removed 9)
123548/679514 => 12348/67914 (removed 5)
123574/325786 => 13574/35786 (removed 2)
6-digit fractions found:9778, omitted {230,256,921,186,317,751,262,205,6650}

"10 minutes and 13s"
```

## Python

`def indexOf(haystack, needle):    idx = 0    for straw in haystack:        if straw == needle:            return idx        else:            idx += 1    return -1 def getDigits(n, le, digits):    while n > 0:        r = n % 10        if r == 0 or indexOf(digits, r) >= 0:            return False        le -= 1        digits[le] = r        n = int(n / 10)    return True def removeDigit(digits, le, idx):    pows = [1, 10, 100, 1000, 10000]    sum = 0    pow = pows[le - 2]    i = 0    while i < le:        if i == idx:            i += 1            continue        sum = sum + digits[i] * pow        pow = int(pow / 10)        i += 1    return sum def main():    lims = [ [ 12, 97 ], [ 123, 986 ], [ 1234, 9875 ], [ 12345, 98764 ] ]    count = [0 for i in range(5)]    omitted = [[0 for i in range(10)] for j in range(5)]     i = 0    while i < len(lims):        n = lims[i]        while n < lims[i]:            nDigits = [0 for k in range(i + 2)]            nOk = getDigits(n, i + 2, nDigits)            if not nOk:                n += 1                continue            d = n + 1            while d <= lims[i] + 1:                dDigits = [0 for k in range(i + 2)]                dOk = getDigits(d, i + 2, dDigits)                if not dOk:                    d += 1                    continue                nix = 0                while nix < len(nDigits):                    digit = nDigits[nix]                    dix = indexOf(dDigits, digit)                    if dix >= 0:                        rn = removeDigit(nDigits, i + 2, nix)                        rd = removeDigit(dDigits, i + 2, dix)                        if (1.0 * n / d) == (1.0 * rn / rd):                            count[i] += 1                            omitted[i][digit] += 1                            if count[i] <= 12:                                print "%d/%d = %d/%d by omitting %d's" % (n, d, rn, rd, digit)                    nix += 1                d += 1            n += 1        print        i += 1     i = 2    while i <= 5:        print "There are %d %d-digit fractions of which:" % (count[i - 2], i)        j = 1        while j <= 9:            if omitted[i - 2][j] == 0:                j += 1                continue            print "%6s have %d's omitted" % (omitted[i - 2][j], j)            j += 1        print        i += 1    return None main()`
Output:
```16/64 = 1/4 by omitting 6's
19/95 = 1/5 by omitting 9's
26/65 = 2/5 by omitting 6's
49/98 = 4/8 by omitting 9's

132/231 = 12/21 by omitting 3's
134/536 = 14/56 by omitting 3's
134/938 = 14/98 by omitting 3's
136/238 = 16/28 by omitting 3's
138/345 = 18/45 by omitting 3's
139/695 = 13/65 by omitting 9's
143/341 = 13/31 by omitting 4's
146/365 = 14/35 by omitting 6's
149/298 = 14/28 by omitting 9's
149/596 = 14/56 by omitting 9's
149/894 = 14/84 by omitting 9's
154/253 = 14/23 by omitting 5's

1234/4936 = 124/496 by omitting 3's
1239/6195 = 123/615 by omitting 9's
1246/3649 = 126/369 by omitting 4's
1249/2498 = 124/248 by omitting 9's
1259/6295 = 125/625 by omitting 9's
1279/6395 = 127/635 by omitting 9's
1283/5132 = 128/512 by omitting 3's
1297/2594 = 127/254 by omitting 9's
1297/3891 = 127/381 by omitting 9's
1298/2596 = 128/256 by omitting 9's
1298/3894 = 128/384 by omitting 9's
1298/5192 = 128/512 by omitting 9's

12349/24698 = 1234/2468 by omitting 9's
12356/67958 = 1236/6798 by omitting 5's
12358/14362 = 1258/1462 by omitting 3's
12358/15364 = 1258/1564 by omitting 3's
12358/17368 = 1258/1768 by omitting 3's
12358/19372 = 1258/1972 by omitting 3's
12358/21376 = 1258/2176 by omitting 3's
12358/25384 = 1258/2584 by omitting 3's
12359/61795 = 1235/6175 by omitting 9's
12364/32596 = 1364/3596 by omitting 2's
12379/61895 = 1237/6185 by omitting 9's
12386/32654 = 1386/3654 by omitting 2's

There are 4 2-digit fractions of which:
2 have 6's omitted
2 have 9's omitted

There are 122 3-digit fractions of which:
9 have 3's omitted
1 have 4's omitted
6 have 5's omitted
15 have 6's omitted
16 have 7's omitted
15 have 8's omitted
60 have 9's omitted

There are 660 4-digit fractions of which:
14 have 1's omitted
25 have 2's omitted
92 have 3's omitted
14 have 4's omitted
29 have 5's omitted
63 have 6's omitted
16 have 7's omitted
17 have 8's omitted
390 have 9's omitted

There are 5087 5-digit fractions of which:
75 have 1's omitted
40 have 2's omitted
376 have 3's omitted
78 have 4's omitted
209 have 5's omitted
379 have 6's omitted
591 have 7's omitted
351 have 8's omitted
2988 have 9's omitted```

## Racket

Racket's generator is horribly slow, so I roll my own more efficient generator. Pretty much using continuation-passing style, but then using macro to make it appear that we are writing in the direct style.

`#lang racket (require racket/generator         syntax/parse/define) (define-syntax-parser for**  [(_ [x:id {~datum <-} (e ...)] rst ...) #'(e ... (λ (x) (for** rst ...)))]  [(_ e ...) #'(begin e ...)]) (define (permutations xs n yield #:lower [lower #f])  (let loop ([xs xs] [n n] [acc '()] [lower lower])    (cond      [(= n 0) (yield (reverse acc))]      [else (for ([x (in-list xs)] #:when (or (not lower) (>= x (first lower))))              (loop (remove x xs)                    (sub1 n)                    (cons x acc)                    (and lower (= x (first lower)) (rest lower))))]))) (define (list->number xs) (foldl (λ (e acc) (+ (* 10 acc) e)) 0 xs)) (define (calc n)  (define rng (range 1 10))  (in-generator   (for** [numer <- (permutations rng n)]          [denom <- (permutations rng n #:lower numer)]          (for* (#:when (not (equal? numer denom))                 [crossed (in-list numer)]                 #:when (member crossed denom)                 [numer* (in-value (list->number (remove crossed numer)))]                 [denom* (in-value (list->number (remove crossed denom)))]                 [numer** (in-value (list->number numer))]                 [denom** (in-value (list->number denom))]                 #:when (= (* numer** denom*) (* numer* denom**)))            (yield (list numer** denom** numer* denom* crossed)))))) (define (enumerate n)  (for ([x (calc n)] [i (in-range 12)])    (apply printf "~a/~a = ~a/~a (~a crossed out)\n" x))  (newline)) (define (stats n)  (define digits (make-hash))  (for ([x (calc n)]) (hash-update! digits (last x) add1 0))  (printf "There are ~a ~a-digit fractions of which:\n" (for/sum ([(k v) (in-hash digits)]) v) n)  (for ([digit (in-list (sort (hash->list digits) < #:key car))])    (printf "  The digit ~a was crossed out ~a times\n" (car digit) (cdr digit)))  (newline)) (define (main)  (enumerate 2)  (enumerate 3)  (enumerate 4)  (enumerate 5)  (stats 2)  (stats 3)  (stats 4)  (stats 5)) (main)`
Output:
```16/64 = 1/4 (6 crossed out)
19/95 = 1/5 (9 crossed out)
26/65 = 2/5 (6 crossed out)
49/98 = 4/8 (9 crossed out)

132/231 = 12/21 (3 crossed out)
134/536 = 14/56 (3 crossed out)
134/938 = 14/98 (3 crossed out)
136/238 = 16/28 (3 crossed out)
138/345 = 18/45 (3 crossed out)
139/695 = 13/65 (9 crossed out)
143/341 = 13/31 (4 crossed out)
146/365 = 14/35 (6 crossed out)
149/298 = 14/28 (9 crossed out)
149/596 = 14/56 (9 crossed out)
149/894 = 14/84 (9 crossed out)
154/253 = 14/23 (5 crossed out)

1234/4936 = 124/496 (3 crossed out)
1239/6195 = 123/615 (9 crossed out)
1246/3649 = 126/369 (4 crossed out)
1249/2498 = 124/248 (9 crossed out)
1259/6295 = 125/625 (9 crossed out)
1279/6395 = 127/635 (9 crossed out)
1283/5132 = 128/512 (3 crossed out)
1297/2594 = 127/254 (9 crossed out)
1297/3891 = 127/381 (9 crossed out)
1298/2596 = 128/256 (9 crossed out)
1298/3894 = 128/384 (9 crossed out)
1298/5192 = 128/512 (9 crossed out)

12349/24698 = 1234/2468 (9 crossed out)
12356/67958 = 1236/6798 (5 crossed out)
12358/14362 = 1258/1462 (3 crossed out)
12358/15364 = 1258/1564 (3 crossed out)
12358/17368 = 1258/1768 (3 crossed out)
12358/19372 = 1258/1972 (3 crossed out)
12358/21376 = 1258/2176 (3 crossed out)
12358/25384 = 1258/2584 (3 crossed out)
12359/61795 = 1235/6175 (9 crossed out)
12364/32596 = 1364/3596 (2 crossed out)
12379/61895 = 1237/6185 (9 crossed out)
12386/32654 = 1386/3654 (2 crossed out)

There are 4 2-digit fractions of which:
The digit 6 was crossed out 2 times
The digit 9 was crossed out 2 times

There are 122 3-digit fractions of which:
The digit 3 was crossed out 9 times
The digit 4 was crossed out 1 times
The digit 5 was crossed out 6 times
The digit 6 was crossed out 15 times
The digit 7 was crossed out 16 times
The digit 8 was crossed out 15 times
The digit 9 was crossed out 60 times

There are 660 4-digit fractions of which:
The digit 1 was crossed out 14 times
The digit 2 was crossed out 25 times
The digit 3 was crossed out 92 times
The digit 4 was crossed out 14 times
The digit 5 was crossed out 29 times
The digit 6 was crossed out 63 times
The digit 7 was crossed out 16 times
The digit 8 was crossed out 17 times
The digit 9 was crossed out 390 times

There are 5087 5-digit fractions of which:
The digit 1 was crossed out 75 times
The digit 2 was crossed out 40 times
The digit 3 was crossed out 376 times
The digit 4 was crossed out 78 times
The digit 5 was crossed out 209 times
The digit 6 was crossed out 379 times
The digit 7 was crossed out 591 times
The digit 8 was crossed out 351 times
The digit 9 was crossed out 2988 times
```

## Raku

(formerly Perl 6)

Works with: Rakudo version 2019.07.1
Anomalous Cancellation
`my %reduced;my \$digits = 2..4; for \$digits.map: * - 1 -> \$exp {    my \$start = sum (0..\$exp).map( { 10 ** \$_ * (\$exp - \$_ + 1) });    my \$end   = 10**(\$exp+1) - sum (^\$exp).map( { 10 ** \$_ * (\$exp - \$_) } ) - 1;     (\$start ..^ \$end).race(:8degree, :3batch).map: -> \$den {        next if \$den.contains: '0';        next if \$den.comb.unique <= \$exp;         for \$start ..^ \$den -> \$num {            next if \$num.contains: '0';            next if \$num.comb.unique <= \$exp;             my \$set = (\$den.comb.head(* - 1).Set ∩ \$num.comb.skip(1).Set);            next if \$set.elems < 1;             for \$set.keys {                my \$ne = \$num.trans: \$_ => '', :delete;                my \$de = \$den.trans: \$_ => '', :delete;                if \$ne / \$de == \$num / \$den {                    print "\b" x 40, "\$num/\$den:\$_ => \$ne/\$de";                    %reduced{"\$num/\$den:\$_"} = "\$ne/\$de";                }            }        }    }      print "\b" x 40, ' ' x 40, "\b" x 40;     my \$digit = \$exp +1;    my %d = %reduced.pairs.grep: { .key.chars == (\$digit * 2 + 3) };    say "\n({+%d}) \$digit digit reduceable fractions:";    for 1..9 {        my \$cnt = +%d.pairs.grep( *.key.contains: ":\$_" );        next unless \$cnt;        say "  \$cnt with removed \$_";    }    say "\n  12 Random (or all, if less) \$digit digit reduceable fractions:";    say "    {.key.substr(0, \$digit * 2 + 1)} => {.value} removed {.key.substr(* - 1)}"      for %d.pairs.pick(12).sort;}`
Sample output:
```(4) 2 digit reduceable fractions:
2 with removed 6
2 with removed 9

12 Random (or all, if less) 2 digit reduceable fractions:
16/64 => 1/4 removed 6
19/95 => 1/5 removed 9
26/65 => 2/5 removed 6
49/98 => 4/8 removed 9

(122) 3 digit reduceable fractions:
9 with removed 3
1 with removed 4
6 with removed 5
15 with removed 6
16 with removed 7
15 with removed 8
60 with removed 9

12 Random (or all, if less) 3 digit reduceable fractions:
149/298 => 14/28 removed 9
154/352 => 14/32 removed 5
165/264 => 15/24 removed 6
176/275 => 16/25 removed 7
187/286 => 17/26 removed 8
194/291 => 14/21 removed 9
286/385 => 26/35 removed 8
286/682 => 26/62 removed 8
374/572 => 34/52 removed 7
473/572 => 43/52 removed 7
492/984 => 42/84 removed 9
594/693 => 54/63 removed 9

(660) 4 digit reduceable fractions:
14 with removed 1
25 with removed 2
92 with removed 3
14 with removed 4
29 with removed 5
63 with removed 6
16 with removed 7
17 with removed 8
390 with removed 9

12 Random (or all, if less) 4 digit reduceable fractions:
1348/4381 => 148/481 removed 3
1598/3196 => 158/316 removed 9
1783/7132 => 178/712 removed 3
1978/5934 => 178/534 removed 9
2971/5942 => 271/542 removed 9
2974/5948 => 274/548 removed 9
3584/4592 => 384/492 removed 5
3791/5798 => 391/598 removed 7
3968/7936 => 368/736 removed 9
4329/9324 => 429/924 removed 3
4936/9872 => 436/872 removed 9
6327/8325 => 627/825 removed 3```

## REXX

`/*REXX pgm reduces fractions by "crossing out" matching digits in nominator&denominator.*/parse arg high show .                            /*obtain optional arguments from the CL*/if high=='' | high==","  then high=  4           /*Not specified?  Then use the default.*/if show=='' | show==","  then show= 12           /* "      "         "   "   "     "    */say center(' some samples of reduced fractions by crossing out digits ', 79, "═")\$.=0                                             /*placeholder array for counts; init. 0*/      do L=2  to high;      say                  /*do 2-dig fractions to HIGH-dig fract.*/                            lim= 10**L - 1       /*calculate the upper limit just once. */         do n=10**(L-1)  to lim                  /*generate some  N  digit  fractions.  */         if pos(0, n) \==0  then iterate         /*Does  it  have a zero?  Then skip it.*/         if hasDup(n)       then iterate         /*  "    "    "  " dup?     "    "   " */             do d=n+1     to lim                           /*only process like-sized #'s */            if pos(0, d)\==0         then iterate         /*Have a zero? Then skip it.  */            if verify(d, n, 'M')==0  then iterate         /*No digs in common?  Skip it.*/            if hasDup(d)             then iterate         /*Any digs are dups?    "   " */            q= n/d                                        /*compute quotient just once. */                  do e=1  for L;     xo= substr(n, e, 1)  /*try crossing out each digit.*/                  nn= space( translate(n, , xo), 0)       /*elide from the numerator.   */                  dd= space( translate(d, , xo), 0)       /*  "     "   "  denominator. */                  if nn/dd \== q  then iterate            /*Not the same quotient? Skip.*/                  \$.L=    \$.L    + 1                      /*Eureka!   We found one.     */                  \$.L.xo= \$.L.xo + 1                      /*count the silly reduction.  */                  if \$.L>show  then iterate               /*Too many found?  Don't show.*/                  say center(n'/'d   " = "   nn'/'dd  "  by crossing out the" xo"'s.", 79)                  end   /*e*/            end         /*d*/         end            /*n*/      end               /*L*/say;                  @with= ' with crossed-out' /* [↓]  show counts for any reductions.*/      do k=1  for 9                              /*traipse through each cross─out digit.*/      if \$.k==0  then iterate                    /*Is this a zero count?  Then skip it. */      say;    say center('There are '     \$.k     " "k'-digit fractions.', 79, "═")                          @for= '          For ' /*literal for SAY indentation (below). */         do #=1  for 9;   if \$.k.#==0  then iterate         say @for    k"-digit fractions, there are "    right(\$.k.#, k-1)   @with   #"'s."         end   /*#*/      end      /*k*/exit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/hasDup: parse arg x;          /* if L<2  then return 0 */           /*L will never be 1.*/           do i=1  for L-1;      if pos(substr(x,i,1), substr(x,i+1)) \== 0  then return 1           end   /*i*/;                                                           return 0`
output   when using the input of:     5   12
```══════════ some samples of reduced fractions by crossing out digits ═══════════

16/64  =  1/4   by crossing out the 6's.
19/95  =  1/5   by crossing out the 9's.
26/65  =  2/5   by crossing out the 6's.
49/98  =  4/8   by crossing out the 9's.

132/231  =  12/21   by crossing out the 3's.
134/536  =  14/56   by crossing out the 3's.
134/938  =  14/98   by crossing out the 3's.
136/238  =  16/28   by crossing out the 3's.
138/345  =  18/45   by crossing out the 3's.
139/695  =  13/65   by crossing out the 9's.
143/341  =  13/31   by crossing out the 4's.
146/365  =  14/35   by crossing out the 6's.
149/298  =  14/28   by crossing out the 9's.
149/596  =  14/56   by crossing out the 9's.
149/894  =  14/84   by crossing out the 9's.
154/253  =  14/23   by crossing out the 5's.

1234/4936  =  124/496   by crossing out the 3's.
1239/6195  =  123/615   by crossing out the 9's.
1246/3649  =  126/369   by crossing out the 4's.
1249/2498  =  124/248   by crossing out the 9's.
1259/6295  =  125/625   by crossing out the 9's.
1279/6395  =  127/635   by crossing out the 9's.
1283/5132  =  128/512   by crossing out the 3's.
1297/2594  =  127/254   by crossing out the 9's.
1297/3891  =  127/381   by crossing out the 9's.
1298/2596  =  128/256   by crossing out the 9's.
1298/3894  =  128/384   by crossing out the 9's.
1298/5192  =  128/512   by crossing out the 9's.

12349/24698  =  1234/2468   by crossing out the 9's.
12356/67958  =  1236/6798   by crossing out the 5's.
12358/14362  =  1258/1462   by crossing out the 3's.
12358/15364  =  1258/1564   by crossing out the 3's.
12358/17368  =  1258/1768   by crossing out the 3's.
12358/19372  =  1258/1972   by crossing out the 3's.
12358/21376  =  1258/2176   by crossing out the 3's.
12358/25384  =  1258/2584   by crossing out the 3's.
12359/61795  =  1235/6175   by crossing out the 9's.
12364/32596  =  1364/3596   by crossing out the 2's.
12379/61895  =  1237/6185   by crossing out the 9's.
12386/32654  =  1386/3654   by crossing out the 2's.

═══════════════════════There are  4  2-digit fractions.════════════════════════
For  2-digit fractions, there are  2  with crossed-out 6's.
For  2-digit fractions, there are  2  with crossed-out 9's.

══════════════════════There are  122  3-digit fractions.═══════════════════════
For  3-digit fractions, there are   9  with crossed-out 3's.
For  3-digit fractions, there are   1  with crossed-out 4's.
For  3-digit fractions, there are   6  with crossed-out 5's.
For  3-digit fractions, there are  15  with crossed-out 6's.
For  3-digit fractions, there are  16  with crossed-out 7's.
For  3-digit fractions, there are  15  with crossed-out 8's.
For  3-digit fractions, there are  60  with crossed-out 9's.

══════════════════════There are  660  4-digit fractions.═══════════════════════
For  4-digit fractions, there are   14  with crossed-out 1's.
For  4-digit fractions, there are   25  with crossed-out 2's.
For  4-digit fractions, there are   92  with crossed-out 3's.
For  4-digit fractions, there are   14  with crossed-out 4's.
For  4-digit fractions, there are   29  with crossed-out 5's.
For  4-digit fractions, there are   63  with crossed-out 6's.
For  4-digit fractions, there are   16  with crossed-out 7's.
For  4-digit fractions, there are   17  with crossed-out 8's.
For  4-digit fractions, there are  390  with crossed-out 9's.

══════════════════════There are  5087  5-digit fractions.══════════════════════
For  5-digit fractions, there are    75  with crossed-out 1's.
For  5-digit fractions, there are    40  with crossed-out 2's.
For  5-digit fractions, there are   376  with crossed-out 3's.
For  5-digit fractions, there are    78  with crossed-out 4's.
For  5-digit fractions, there are   209  with crossed-out 5's.
For  5-digit fractions, there are   379  with crossed-out 6's.
For  5-digit fractions, there are   591  with crossed-out 7's.
For  5-digit fractions, there are   351  with crossed-out 8's.
For  5-digit fractions, there are  2988  with crossed-out 9's.
```

## Visual Basic .NET

Translation of: C#
`Module Module1     Function IndexOf(n As Integer, s As Integer()) As Integer        For ii = 1 To s.Length            Dim i = ii - 1            If s(i) = n Then                Return i            End If        Next        Return -1    End Function     Function GetDigits(n As Integer, le As Integer, digits As Integer()) As Boolean        While n > 0            Dim r = n Mod 10            If r = 0 OrElse IndexOf(r, digits) >= 0 Then                Return False            End If            le -= 1            digits(le) = r            n \= 10        End While        Return True    End Function     Function RemoveDigit(digits As Integer(), le As Integer, idx As Integer) As Integer        Dim pows = {1, 10, 100, 1000, 10000}        Dim sum = 0        Dim pow = pows(le - 2)        For ii = 1 To le            Dim i = ii - 1            If i = idx Then                Continue For            End If            sum += digits(i) * pow            pow \= 10        Next        Return sum    End Function     Sub Main()        Dim lims = {{12, 97}, {123, 986}, {1234, 9875}, {12345, 98764}}        Dim count(5) As Integer        Dim omitted(5, 10) As Integer        Dim upperBound = lims.GetLength(0)        For ii = 1 To upperBound            Dim i = ii - 1            Dim nDigits(i + 2 - 1) As Integer            Dim dDigits(i + 2 - 1) As Integer            Dim blank(i + 2 - 1) As Integer            For n = lims(i, 0) To lims(i, 1)                blank.CopyTo(nDigits, 0)                Dim nOk = GetDigits(n, i + 2, nDigits)                If Not nOk Then                    Continue For                End If                For d = n + 1 To lims(i, 1) + 1                    blank.CopyTo(dDigits, 0)                    Dim dOk = GetDigits(d, i + 2, dDigits)                    If Not dOk Then                        Continue For                    End If                    For nixt = 1 To nDigits.Length                        Dim nix = nixt - 1                        Dim digit = nDigits(nix)                        Dim dix = IndexOf(digit, dDigits)                        If dix >= 0 Then                            Dim rn = RemoveDigit(nDigits, i + 2, nix)                            Dim rd = RemoveDigit(dDigits, i + 2, dix)                            If (n / d) = (rn / rd) Then                                count(i) += 1                                omitted(i, digit) += 1                                If count(i) <= 12 Then                                    Console.WriteLine("{0}/{1} = {2}/{3} by omitting {4}'s", n, d, rn, rd, digit)                                End If                            End If                        End If                    Next                Next            Next            Console.WriteLine()        Next         For i = 2 To 5            Console.WriteLine("There are {0} {1}-digit fractions of which:", count(i - 2), i)            For j = 1 To 9                If omitted(i - 2, j) = 0 Then                    Continue For                End If                Console.WriteLine("{0,6} have {1}'s omitted", omitted(i - 2, j), j)            Next            Console.WriteLine()        Next    End Sub End Module`
Output:
```16/64 = 1/4 by omitting 6's
19/95 = 1/5 by omitting 9's
26/65 = 2/5 by omitting 6's
49/98 = 4/8 by omitting 9's

132/231 = 12/21 by omitting 3's
134/536 = 14/56 by omitting 3's
134/938 = 14/98 by omitting 3's
136/238 = 16/28 by omitting 3's
138/345 = 18/45 by omitting 3's
139/695 = 13/65 by omitting 9's
143/341 = 13/31 by omitting 4's
146/365 = 14/35 by omitting 6's
149/298 = 14/28 by omitting 9's
149/596 = 14/56 by omitting 9's
149/894 = 14/84 by omitting 9's
154/253 = 14/23 by omitting 5's

1234/4936 = 124/496 by omitting 3's
1239/6195 = 123/615 by omitting 9's
1246/3649 = 126/369 by omitting 4's
1249/2498 = 124/248 by omitting 9's
1259/6295 = 125/625 by omitting 9's
1279/6395 = 127/635 by omitting 9's
1283/5132 = 128/512 by omitting 3's
1297/2594 = 127/254 by omitting 9's
1297/3891 = 127/381 by omitting 9's
1298/2596 = 128/256 by omitting 9's
1298/3894 = 128/384 by omitting 9's
1298/5192 = 128/512 by omitting 9's

12349/24698 = 1234/2468 by omitting 9's
12356/67958 = 1236/6798 by omitting 5's
12358/14362 = 1258/1462 by omitting 3's
12358/15364 = 1258/1564 by omitting 3's
12358/17368 = 1258/1768 by omitting 3's
12358/19372 = 1258/1972 by omitting 3's
12358/21376 = 1258/2176 by omitting 3's
12358/25384 = 1258/2584 by omitting 3's
12359/61795 = 1235/6175 by omitting 9's
12364/32596 = 1364/3596 by omitting 2's
12379/61895 = 1237/6185 by omitting 9's
12386/32654 = 1386/3654 by omitting 2's

There are 4 2-digit fractions of which:
2 have 6's omitted
2 have 9's omitted

There are 122 3-digit fractions of which:
9 have 3's omitted
1 have 4's omitted
6 have 5's omitted
15 have 6's omitted
16 have 7's omitted
15 have 8's omitted
60 have 9's omitted

There are 660 4-digit fractions of which:
14 have 1's omitted
25 have 2's omitted
92 have 3's omitted
14 have 4's omitted
29 have 5's omitted
63 have 6's omitted
16 have 7's omitted
17 have 8's omitted
390 have 9's omitted

There are 5087 5-digit fractions of which:
75 have 1's omitted
40 have 2's omitted
376 have 3's omitted
78 have 4's omitted
209 have 5's omitted
379 have 6's omitted
591 have 7's omitted
351 have 8's omitted
2988 have 9's omitted```

## zkl

Translation of: Phix
`fcn toInt(digits,remove_digit=0){   if(remove_digit!=0) digits=digits.copy().del(digits.index(remove_digit));   digits.reduce(fcn(s,d){ s*10 + d });}fcn nDigits(n){    //-- generate numbers with unique digits efficiently     //-- and store them in an array for multiple re-use,    //-- along with an array of the removed-digit values.    res,digits := List(), n.pump(List(),'+(1));  // 1,2,3,4..n    used       := List.createLong(n,1).extend(List.createLong(9-n,0));    while(True){       nine:=List.createLong(9,0);       foreach i in (used.len()){ if(used[i]) nine[i]=toInt(digits,i+1) }       res.append(T(toInt(digits),nine));       found:=False;       foreach i in ([n-1..0, -1]){          d:=digits[i];	  if(not used[d-1]) println("ack!");	  used[d-1]=0;	  foreach j in ([d..8]){	     if(not used[j]){		used[j]=1;		digits[i]=j+1;		foreach k in ([i+1..n-1]){		   digits[k] = used.find(0) + 1;		   used[digits[k] - 1]=1;		}		found=True;		break;	     }	  }	  if(found) break;       }//foreach i       if(not found) break;    }//while    res} foreach n in ([2..5]){   rs,rsz,count,omitted := nDigits(n),rs.len()-1, 0, List.createLong(9,0);   foreach i in (rsz){      xn,rn := rs[i];      foreach j in ([i+1..rsz]){         xd,rd := rs[j];	 foreach k in ([0..8]){	    yn,yd := rn[k],rd[k];	    if(yn!=0 and yd!=0 and 	         xn.toFloat()/xd.toFloat() == yn.toFloat()/yd.toFloat()){	       count+=1;	       omitted[k]+=1;	       if(count<=12)		  println("%d/%d --> %d/%d (removed %d)".fmt(xn,xd,yn,yd,k+1));	    }	 }      }   }   println("%d-digit fractions found: %d, omitted %s\n"      .fmt(n,count,omitted.concat(",")));}`
Output:
```16/64 --> 1/4 (removed 6)
19/95 --> 1/5 (removed 9)
26/65 --> 2/5 (removed 6)
49/98 --> 4/8 (removed 9)
2-digit fractions found: 4, omitted 0,0,0,0,0,2,0,0,2

132/231 --> 12/21 (removed 3)
134/536 --> 14/56 (removed 3)
134/938 --> 14/98 (removed 3)
136/238 --> 16/28 (removed 3)
138/345 --> 18/45 (removed 3)
139/695 --> 13/65 (removed 9)
143/341 --> 13/31 (removed 4)
146/365 --> 14/35 (removed 6)
149/298 --> 14/28 (removed 9)
149/596 --> 14/56 (removed 9)
149/894 --> 14/84 (removed 9)
154/253 --> 14/23 (removed 5)
3-digit fractions found: 122, omitted 0,0,9,1,6,15,16,15,60

1234/4936 --> 124/496 (removed 3)
1239/6195 --> 123/615 (removed 9)
1246/3649 --> 126/369 (removed 4)
1249/2498 --> 124/248 (removed 9)
1259/6295 --> 125/625 (removed 9)
1279/6395 --> 127/635 (removed 9)
1283/5132 --> 128/512 (removed 3)
1297/2594 --> 127/254 (removed 9)
1297/3891 --> 127/381 (removed 9)
1298/2596 --> 128/256 (removed 9)
1298/3894 --> 128/384 (removed 9)
1298/5192 --> 128/512 (removed 9)
4-digit fractions found: 660, omitted 14,25,92,14,29,63,16,17,390

12349/24698 --> 1234/2468 (removed 9)
12356/67958 --> 1236/6798 (removed 5)
12358/14362 --> 1258/1462 (removed 3)
12358/15364 --> 1258/1564 (removed 3)
12358/17368 --> 1258/1768 (removed 3)
12358/19372 --> 1258/1972 (removed 3)
12358/21376 --> 1258/2176 (removed 3)
12358/25384 --> 1258/2584 (removed 3)
12359/61795 --> 1235/6175 (removed 9)
12364/32596 --> 1364/3596 (removed 2)
12379/61895 --> 1237/6185 (removed 9)
12386/32654 --> 1386/3654 (removed 2)
5-digit fractions found: 5087, omitted 75,40,376,78,209,379,591,351,2988
```