Largest palindrome product

Largest palindrome product is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Task

Task description is taken from Project Euler (https://projecteuler.net/problem=4)
A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.
Find the largest palindrome made from the product of two 3-digit numbers.

Stretch Goal

Find the largest palindrome made from the product of two n-digit numbers, where n ranges from 4 to 7.

Extended Stretch Goal

Find the largest palindrome made from the product of two n-digit numbers, where n ranges beyond 7,

11l

Translation of: Wren

F reverse(=n)
   V r = Int64(0)
   L n > 0
      r = n % 10 + r * 10
      n I/= 10
   R r

V po = Int64(10)
L(n) 2..7
   V low = po * 9
   po *= 10
   V high = po - 1
   V nextN = 0B
   L(i) (high .. low).step(-1)
      V j = reverse(i)
      V p = i * po + j
      V k = high
      L k > low
         I k % 10 != 5
            V l = p I/ k
            I l > high
               L.break
            I p % k == 0
               print(‘Largest palindromic product of two ’n‘-digit integers: ’k‘ x ’l‘ = ’p)
               nextN = 1B
               L.break
         k -= 2
      I nextN
         L.break

Output:

Largest palindromic product of two 2-digit integers: 99 x 91 = 9009
Largest palindromic product of two 3-digit integers: 993 x 913 = 906609
Largest palindromic product of two 4-digit integers: 9999 x 9901 = 99000099
Largest palindromic product of two 5-digit integers: 99979 x 99681 = 9966006699
Largest palindromic product of two 6-digit integers: 999999 x 999001 = 999000000999
Largest palindromic product of two 7-digit integers: 9998017 x 9997647 = 99956644665999

ALGOL 68

Translation of: Wren

BEGIN # find the highest palindromic multiple of various sizes of numbers #
    # returns n with the digits reversed #
    PROC reverse = ( LONG INT v )LONG INT:
         BEGIN
            LONG INT r := 0 ;
            LONG INT n := v;
            WHILE n > 0 DO
                r *:= 10; r +:= n MOD 10;
                n OVERAB 10
            OD;
            r
         END # reverse # ;
 
    LONG INT pow := 10;
    FOR n FROM 2 TO 7 DO
        LONG INT low := pow * 9;
        pow *:= 10;
        LONG INT high := pow - 1;
        print( ( "Largest palindromic product of two ", whole( n, 0 ), "-digit integers: " ) );
        BOOL next n := FALSE;
        LONG INT i := high + 1;
        WHILE i -:= 1;
              i >= low AND NOT next n
        DO
            LONG INT j = reverse( i );
            LONG INT p = ( i * pow ) + j;
            # k can't be even nor end in 5 to produce a product ending in 9 #
            LONG INT k := high + 2;
            WHILE k -:= 2;
                  IF   k < low
                  THEN FALSE
                  ELIF k MOD 10 = 5
                  THEN TRUE
                  ELIF LONG INT l = p OVER k;
                                l > high
                  THEN FALSE
                  ELIF p MOD k = 0 THEN
                       print( ( whole( k, 0 ), " x ", whole( l, 0 ), " = ", whole( p, 0 ), newline ) );
                       next n := TRUE;
                       FALSE
                  ELSE TRUE
                  FI
            DO SKIP OD
        OD
    OD
END

Output:

Largest palindromic product of two 2-digit integers: 99 x 91 = 9009
Largest palindromic product of two 3-digit integers: 993 x 913 = 906609
Largest palindromic product of two 4-digit integers: 9999 x 9901 = 99000099
Largest palindromic product of two 5-digit integers: 99979 x 99681 = 9966006699
Largest palindromic product of two 6-digit integers: 999999 x 999001 = 999000000999
Largest palindromic product of two 7-digit integers: 9998017 x 9997647 = 99956644665999

Translation of: Ring

Also showing the maximum for 2 and 4 .. 7 digit numbers. Tests for a better product before testing for palindromicity.

BEGIN # find the highest palindromic multiple of various sizes of numbers #
    PROC is pal = ( LONG INT n )BOOL:
         BEGIN
             STRING x       = whole( n, 0 );
             INT    l      := UPB x + 1;
             BOOL   result := TRUE;
             FOR i FROM LWB x WHILE i < l AND result DO
                 l     -:= 1;
                 result := x[ i ] = x[ l ]
             OD;
             result
          END # is pal # ;

    # maximum 2 digit number #
    LONG INT max := 99;
    # both factors must be >= 10for a 4 digit product #
    LONG INT limit start := 10;
    FOR w FROM 2 TO 7 DO
        LONG INT best prod   := 0;
        # one factor must be divisible by 11 #
        LONG INT limit end = 11 * ( max OVER 11 );
        LONG INT second   := limit start;
        LONG INT first    := 1;
        # loop from hi to low to find the best result in the fewest steps #
        LONG INT n        := limit end + 11;
        WHILE n -:= 11;
              n >= limit start
        DO
            # with n falling, the lower limit of m can rise with #
            # the best-found-so-far second number. Doing this #
            # lowers the iteration count by a lot. #
            LONG INT m := max + 2;
            WHILE m -:= 2;
                  IF   m < second
                  THEN FALSE
                  ELIF LONG INT prod = n * m;
                       best prod > prod
                  THEN FALSE
                  ELIF NOT is pal( prod )
                  THEN TRUE
                  ELSE # maintain the best-found-so-far result #
                       first     := n;
                       second    := m;
                       best prod := prod;
                       TRUE
                  FI
            DO SKIP OD
        OD; 
        print( ( "Largest palindromic product of two ", whole( w, 0 )
               , "-digit numbers: ", whole( first, 0 ), " * ", whole( second, 0 )
               , " = ", whole( best prod, 0 )
               , newline
               )
             );
        max *:= 10;
        max +:= 9;
        limit start *:= 10
    OD
END

Output:

Largest palindromic product of two 2-digit numbers: 99 * 91 = 9009
Largest palindromic product of two 3-digit numbers: 913 * 993 = 906609
Largest palindromic product of two 4-digit numbers: 9999 * 9901 = 99000099
Largest palindromic product of two 5-digit numbers: 99979 * 99681 = 9966006699
Largest palindromic product of two 6-digit numbers: 999999 * 999001 = 999000000999
Largest palindromic product of two 7-digit numbers: 9997647 * 9998017 = 99956644665999

AWK

# syntax: GAWK -f LARGEST_PALINDROME_PRODUCT.AWK
BEGIN {
    main(9)
    main(99)
    main(999)
    main(9999)
    exit(0)
}
function main(n,  i,j,max_i,max_j,max_product,product) {
    for (i=1; i<=n; i++) {
      for (j=1; j<=n; j++) {
        product = i * j
        if (product > max_product) {
          if (product ~ /^9/ && product ~ /9$/) {
            if (product == reverse(product)) {
              max_product = product
              max_i = i
              max_j = j
            }
          }
        }
      }
    }
    printf("%1d: %4s * %-4s = %d\n",length(n),max_i,max_j,max_product)
}
function reverse(str,  i,rts) {
    for (i=length(str); i>=1; i--) {
      rts = rts substr(str,i,1)
    }
    return(rts)
}

Output:

1:    1 * 9    = 9
2:   91 * 99   = 9009
3:  913 * 993  = 906609
4: 9901 * 9999 = 99000099

Ksh

#!/bin/ksh

# Largest palindrome product of two 3-digit numbers

#	# Variables:
#
typeset -si MINFACT=913		# From 'Paper & Pencil' solution
typeset -si MAXFACT=999

#	# Functions:
#

#	# Function _ispalindrome(n) - return 1 for palindromic number
#
function _ispalindrome {
	typeset _n ; integer _n="$1"

	(( _n != $(_flipit ${_n}) )) && return 0
	return 1
}

#	# Function _flipit(string) - return flipped string
#
function _flipit {
	typeset _buf ; _buf="$1"
	typeset _tmp ; unset _tmp
	typeset _i ; typeset -si _i

	for (( _i=$(( ${#_buf}-1 )); _i>=0; _i-- )); do
		_tmp="${_tmp}${_buf:${_i}:1}"
	done
	echo "${_tmp}"
}

 ######
# main #
 ######

integer prod MAXPPROD=0
for (( i=MINFACT; i<=MAXFACT; i++)); do
	for (( j=MINFACT; j<=MAXFACT; j++)); do
		(( prod = i * j ))
		_ispalindrome ${prod}
		(( $? )) && (( prod > MAXPPROD )) && MAXPPROD=${prod}
	done
done

print "Largest palindrome product of two 3-digit factors = ${MAXPPROD}"

Output:

Largest palindrome product of two 3-digit factors = 906609

C#

Main Task

Translation of: Paper & Pencil

using System;
class Program {

  static bool isPal(int n) {
    int rev = 0, lr = -1, rem;
    while (n > rev) {
      n = Math.DivRem(n, 10, out rem);
      if (lr < 0 && rem == 0) return false;
      lr = rev; rev = 10 * rev + rem;
      if (n == rev || n == lr) return true;
    } return false; }

  static void Main(string[] args) {
    var sw = System.Diagnostics.Stopwatch.StartNew();
    int x = 900009, y = (int)Math.Sqrt(x), y10, max = 999, max9 = max - 9, z, p, bp = x, ld, c;
    var a = new int[]{ 0,9,0,3,0,0,0,7,0,1 }; string bs = "";
    y /= 11;
    if ((y & 1) == 0) y--;
    if (y % 5 == 0) y -= 2;
    y *= 11;
    while (y <= max) {
      c = 0;
      y10 = y * 10;
      z = max9 + a[ld = y % 10];
      p = y * z;
      while (p >= bp) {
        if (isPal(p)) {
          if (p > bp) bp = p;
          bs = string.Format("{0} x {1} = {2}", y, z - c, bp);
        }
        p -= y10; c += 10;
      }
      y += ld == 3 ? 44 : 22;
    }
    sw.Stop();
    Console.Write("{0} {1} μs", bs, sw.Elapsed.TotalMilliseconds * 1000.0);
  }
}

Output @ Tio.run:

913 x 993 = 906609 245.2 μs

Stretch

using System;

class Program {

    static bool isPal(long n) {
        long rev = 0, lr = -1, rem;
        while (n > rev) {
            n = Math.DivRem(n, 10, out rem);
            if (lr < 0 && rem == 0) return false;
            lr = rev; rev = 10 * rev + rem;
            if (n == rev || n == lr) return true;
        } return false; }

    static void doOne(int n) {
        int ld, c; string bs = "";
        string sx = "9" + new string('0', (n - 1) << 1) + "9", sm = new string('9', n);
        long x = long.Parse(sx), y = (long)Math.Sqrt(x), oy, max = long.Parse(sm), max9 = max - 9, z, yy, p, bp = x;
        var a = new long[] { 0, 9, 0, 3, 0, 0, 0, 7, 0, 1 };
        y /= 11;
        if ((y & 1) == 0) y--;
        if (y % 5 == 0) y -= 2;
        y *= 11; oy = y;
        while (y <= max) y += 22; y -= 22;
        while (y >= oy) {
            c = 0;
            yy = y * 10;
            z = max9 + a[ld = (int)(y % 10)];
            //Console.WriteLine("y,z: {0},{1}", y, z);
            p = y * z;
            while (p >= bp) {
                if (isPal(p)) {
                    if (p > bp) bp = p;
                    bs = string.Format(" {0,2} {1,10} x {2,-10} = {3}{4}", n, y, z - c, new string(' ', 10 - n), bp); }
                p -= yy; c += 10; }
            y -= ld == 7 ? 44 : 22; }
        Console.WriteLine(bs); }

    static void Main(string[] args) {
        Console.WriteLine("digs    factor   factor            palindrome");
        var sw = System.Diagnostics.Stopwatch.StartNew();
        for (int i = 2, h = 1; i <= 10; h = ++i >> 1) {
            if ((i & 1) == 0) {
                string b = new string('9', i),
                       a = new string('9', h) + new string('0', (h) - 1) + "1",
                       c = new string('9', h) + new string('0', i) + new string('9', h);
                Console.WriteLine(" {0,2} {1,10} x {2,-10} = {3}{4}", i, a, b, new string(' ', 10 - i), c); }
            else doOne(i);
        }
        sw.Stop();
        Console.Write("{0} sec", sw.Elapsed.TotalSeconds);
    }
}

Output @ Tio.run:

Showing results for 2 through 10 digit factors.

digs    factor   factor            palindrome
  2         91 x 99         =         9009
  3        913 x 993        =        906609
  4       9901 x 9999       =       99000099
  5      99979 x 99681      =      9966006699
  6     999001 x 999999     =     999000000999
  7    9997647 x 9998017    =    99956644665999
  8   99990001 x 99999999   =   9999000000009999
  9  999920317 x 999980347  =  999900665566009999
 10 9999900001 x 9999999999 = 99999000000000099999
2.1622142 sec

Wow! how did that go so fast? The results for the even-number-of-digit factors were manufactured by string manipulation instead of calculation (since the pattern was obvious). This algorithm can easily be adapted to BigIntegers for higher n-digit factors, but the execution time is unspectacular.

F#

// Largest palindrome product. Nigel Galloway: November 3rd., 2021
let fN g=let rec fN g=[yield g%10; if g>=10 then yield! fN(g/10)] in let n=fN g in n=List.rev n
printfn "%d" ([for n in 100..999 do for g in n..999->n*g]|>List.filter fN|>List.max)

Output:

906609

FreeBASIC

Version 1

function make_pal( n as ulongint ) as ulongint
    'turn a number into a palindrom with twice as many digits
    dim as string ns, ret
    ns = str(n) : ret = ns
    for i as uinteger = len(ns) to 1 step -1
       ret += mid(ns, i, 1)
    next i
    return val(ret)
end function

function has_dig( n as ulongint, d as uinteger ) as boolean
    'does the number n have d decimal digits?
    if 10^(d-1)<=n and n<10^d then return true else return false
end function

dim as integer np

for d as uinteger = 2 to 7
    for n as ulongint = 10^d - 1 to 10^(d-1) step -1 'count down from 999...
                                                     'since the first good number we encounter
                                                     'must be the highest
        np = make_pal( n )                           'produce a 2d-digit palindrome from it
        for f as ulongint = 10^d - 1 to 10^(d-1) step -1   'look for highest d-digit factor
            if np mod f = 0 then
                if has_dig( np/f, d ) then           'if np/f also has d digits we are done :)
                    print f;" *";np/f;" =";np
                    goto nextd
                end if
            end if
        next f
    next n
    nextd:                                           'yes, I used a goto. sue me.
next d

Output:

99 * 91 = 9009
993 * 913 = 906609
9999 * 9901 = 99000099
99979 * 99681 = 9966006699
999999 * 999001 = 999000000999
9998017 * 9997647 = 99956644665999

Version 2

This version is based on Version 1 with only a few changes and some extra code based on the fact that one divisor can be divided by 11, this speeds it even more up and a option for using goto, exit or continue. highest n is 9, (highest possible for unsigned 64bit integers).

' version 07-10-2021
' compile with: fbc -s console

' Now you can choice, no speed changes for all 3
' 1: use goto
' 2: use exit
' 3: use continue
#Define Option_ 1  ' set option_ to 1, 2 or 3. for all other value's uses 1

Function make_pal( n As UInteger ) As ULongInt
    'turn a number into a palindrom with twice as many digits
    Dim As String ns = Str(n), ret = ns
    For i As UInteger = Len(ns) To 1 Step -1
        ret += Mid(ns, i, 1)
    Next i
    Return ValULng(ret)
End Function

Dim As ULongInt np, tmp
Dim As Double t1 =Timer
For d As UInteger = 2 To 9
    For n As UInteger = 10^d -2 To 10^(d -1) Step -1
        np = make_pal( n )
        tmp = Sqr(np)
        tmp = tmp - (10^d - 1 - tmp)
        tmp = tmp - tmp Mod 11
        If (tmp And 1) = 0 Then tmp = tmp + 11
        For f As UInteger = tmp To 10^d -1  Step 22
            If np Mod f = 0 Then
                If np \ f > (10^d) Then Continue For
                Print f; " * "; np \ f; " = "; np
                #If (option_ = 2)
                    Exit For, For
                #ElseIf (option_ = 3)
                    Continue For, For, For
                #Else
                    GoTo nextd
                #EndIf
            End If
        Next f
    Next n
    #If (option_ <> 2 Or option_ <> 3)
        nextd:
    #EndIf
Next d

Print Timer-t1
' empty keyboard buffer
While InKey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End

Output:

99 * 91 = 9009
993 * 913 = 906609
9999 * 9901 = 99000099
99979 * 99681 = 9966006699
999999 * 999001 = 999000000999
9998017 * 9997647 = 99956644665999
99999999 * 99990001 = 9999000000009999
999980347 * 999920317 = 999900665566009999

Go

Translation of: Wren

18 digit integers are within the range of Go's uint64 type though finding the result for 9-digit number products takes a while - around 15 seconds on my machine.

package main

import "fmt"

func reverse(n uint64) uint64 {
    r := uint64(0)
    for n > 0 {
        r = n%10 + r*10
        n /= 10
    }
    return r
}

func main() {
    pow := uint64(10)
nextN:
    for n := 2; n < 10; n++ {
        low := pow * 9
        pow *= 10
        high := pow - 1
        fmt.Printf("Largest palindromic product of two %d-digit integers: ", n)
        for i := high; i >= low; i-- {
            j := reverse(i)
            p := i*pow + j
            // k can't be even nor end in 5 to produce a product ending in 9
            for k := high; k > low; k -= 2 {
                if k % 10 == 5 {
                    continue
                }
                l := p / k
                if l > high {
                    break
                }
                if p%k == 0 {
                    fmt.Printf("%d x %d = %d\n", k, l, p)
                    continue nextN
                }
            }
        }
    }
}

Output:

Largest palindromic product of two 2-digit integers: 99 x 91 = 9009
Largest palindromic product of two 3-digit integers: 993 x 913 = 906609
Largest palindromic product of two 4-digit integers: 9999 x 9901 = 99000099
Largest palindromic product of two 5-digit integers: 99979 x 99681 = 9966006699
Largest palindromic product of two 6-digit integers: 999999 x 999001 = 999000000999
Largest palindromic product of two 7-digit integers: 9998017 x 9997647 = 99956644665999
Largest palindromic product of two 8-digit integers: 99999999 x 99990001 = 9999000000009999
Largest palindromic product of two 9-digit integers: 999980347 x 999920317 = 999900665566009999

jq

Adapted from Wren

Works with: jq

Works with gojq, the Go implementation of jq

def reverseNumber:
  tostring|explode|reverse|implode|tonumber;
 
def task:
  { pow: 10}
  | foreach range(2;8) as $n (.;
      (.pow * 9) as $low
      | .pow *= 10
      | (.pow - 1) as $high
      | .emit = null
      | .nextN = false
      | label $out
      | foreach range($high; $low - 1; -1) as $i (.;
          ($i|reverseNumber) as $j
          | ($i * .pow + $j) as $p
          # k can't be even nor end in 5 to produce a product ending in 9
          | .k = $high
	  | .done = false
          | until(.k <= $low or .done;
              if (.k % 10 != 5)
              then ($p / .k) as $l
              | if $l > $high
	        then .done = true
                elif $p % .k == 0
                then .emit = "Largest palindromic product of two \($n)-digit integers: \(.k) x \($l) = \($p)"
                | .nextN = true
                | .done = true
                else .
		end
	      else .
  	      end
              | .k += -2 )
          | if .nextN then ., break $out else . end;
	  select(.emit) );
    .emit ) ;

task

Output:

Largest palindromic product of two 2-digit integers: 99 x 91 = 9009
Largest palindromic product of two 3-digit integers: 993 x 913 = 906609
Largest palindromic product of two 4-digit integers: 9999 x 9901 = 99000099
Largest palindromic product of two 5-digit integers: 99979 x 99681 = 9966006699
Largest palindromic product of two 6-digit integers: 999999 x 999001 = 999000000999
Largest palindromic product of two 7-digit integers: 9998017 x 9997647 = 99956644665999

Julia

using Primes

function twoprodpal(factorlength)
    maxpal = Int128(10)^(2 * factorlength) - 1
    dig = digits(maxpal)
    halfnum = dig[1:length(dig)÷2]
    while any(halfnum .!= 0)
        prodnum = evalpoly(Int128(10), [reverse(halfnum); halfnum])
        facs = twofac(factorlength, prodnum)
        if !isempty(facs)
            println("For factor length $factorlength, $(facs[1]) * $(facs[2]) = $prodnum")
            break
        end
        halfnum = digits(evalpoly(Int128(10), halfnum) - 1)
    end
end

function twofac(faclength, prodnum)
    f = [one(prodnum)]
    for (p, e) in factor(prodnum)
        f = reduce(vcat, [f * p^j for j in 1:e], init=f)
    end
    possiblefacs = filter(x -> length(string(x)) == faclength, f)
    for i in possiblefacs
        j = prodnum ÷ i
        j ∈ possiblefacs && return sort([i, j])
    end
    return typeof(prodnum)[]
end

@Threads.threads for i in 2:12
    twoprodpal(i)
end

Output:

For factor length 2, 91 * 99 = 9009
For factor length 3, 913 * 993 = 906609
For factor length 4, 9901 * 9999 = 99000099
For factor length 5, 99681 * 99979 = 9966006699
For factor length 6, 999001 * 999999 = 999000000999
For factor length 7, 9997647 * 9998017 = 99956644665999
For factor length 8, 99990001 * 99999999 = 9999000000009999
For factor length 9, 999920317 * 999980347 = 999900665566009999
For factor length 10, 9999986701 * 9999996699 = 99999834000043899999
For factor length 11, 99999943851 * 99999996349 = 9999994020000204999999
For factor length 12, 999999000001 * 999999999999 = 999999000000000000999999

Faster version

Translation of: Python

""" taken from https://leetcode.com/problems/largest-palindrome-product/discuss/150954/Fast-algorithm-by-constrains-on-tail-digits """

const T = [Set([(0, 0)])]

function double(it)
    arr = empty(it)
    for p in it
        push!(arr, p, reverse(p))
    end
    return arr
end

""" Construct a pair of n-digit numbers such that their product ends with 99...9 pattern """
function tails(n)
    if length(T) <= n
        l = Set()
        for i in 0:9, j in i:9
            I = i * 10^(n-1)
            J = j * 10^(n-1)
            it = collect(tails(n - 1))
            I != J && (it = double(it))
            for (t1, t2) in it
                if ((I + t1) * (J + t2) + 1) % 10^n == 0
                    push!(l, (I + t1, J + t2))
                end
            end
        end
        push!(T, l)
    end
    return T[n + 1]
end

""" find the largest palindrome that is a product of n-digit numbers """
function largestpalindrome(n)
    m, tail = 0, n ÷ 2
    head = n - tail
    up = 10^head
    for L in 1 : 9 * 10^(head-1)
        # Consider small shell (up-L)^2 < (up-i)*(up-j) <= (up-L)^2, 1<=i<=L<=j
        m, sol = 0, (0, 0)
        for i in 1:L
            lo = max(Int128(i), Int128(up - (up - L + 1)^2 ÷ (up - i)) + 1)
            hi = Int128(up - (up - L)^2 ÷ (up - i))
            for j in lo:hi
                I = (up - i) * 10^tail
                J = (up - j) * 10^tail
                it = collect(tails(tail))
                I != J && (it = double(it))
                for (t1, t2) in it
                    val = (I + t1) * (J + t2)
                    s = string(val)
                    if s == reverse(s) && val > m
                        sol = (I + t1, J + t2)
                        m = val
                    end
                end
            end
        end
        if m > 0
            println(lpad(n, 2), "    ", lpad(m % 1337, 4), " $sol $(sol[1] * sol[2])")
            return m % 1337
        end
    end
    return 0
end

@time for k in 1:16
    largestpalindrome(k)
end

Output:

 1       9 (9, 1) 9
 2     987 (91, 99) 9009
 3     123 (993, 913) 906609
 4     597 (9901, 9999) 99000099
 5     677 (99979, 99681) 9966006699
 6    1218 (999001, 999999) 999000000999
 7     877 (9998017, 9997647) 99956644665999
 8     475 (99990001, 99999999) 9999000000009999
 9    1226 (999980347, 999920317) 999900665566009999
10     875 (9999986701, 9999996699) 99999834000043899999
11     108 (99999943851, 99999996349) 9999994020000204999999
12     378 (999999000001, 999999999999) 999999000000000000999999
13    1097 (9999999993349, 9999996340851) 99999963342000024336999999
14     959 (99999990000001, 99999999999999) 9999999000000000000009999999
15     465 (999999998341069, 999999975838971) 999999974180040040081479999999
16      51 (9999999900000001, 9999999999999999) 99999999000000000000000099999999
 62.575515 seconds (241.50 M allocations: 16.491 GiB, 25.20% gc time, 0.07% compilation time)

Mathematica / Wolfram Language

palindromeQ[n_] := (* faster than built in test PalindromeQ *)
 Block[{digits = IntegerDigits@n}, digits == Reverse[digits]]

 nextPair[n_] := (* outputs next pair of candidate divisors *)
  Block[{next = 
    NestWhile[# - 11 &, n, ! MemberQ[{1, 3, 7, 9}, Mod[#, 10]] &], 
   len = Last@RealDigits@n},
  {next, 10^len - Switch[Mod[next, 10], 1, 1, 3, 7, 7, 3, 9, 9]}]

search[n_] := 
 Block[{resetLimit = 10^(n - Floor[n/2]) (10^Floor[n/2] - 1), cands},
  cands = 
   Partition[
    Flatten[
     Reap[
      NestWhile[(If[palindromeQ[Times @@ #], Sow[#]]; 
         If[Last@# < resetLimit, 
          nextPair[First@# - 11], # - {0, 10}]) &, 
       nextPair@If[EvenQ@n, 10^n - 1, 10^n - 21], 
       First@# > resetLimit &]]], 2];
  Flatten@cands[[Ordering[Times @@@ cands, -1]]]]

Grid[Join[{{"factors", "largest palindrome"}}, {#, Times @@ #} & /@ 
   Table[search[n], {n, 2, 7}]], Alignment -> {Left, Baseline}]

Output:

factors			largest palindrome
{99,91}			9009
{913,993}       	906609
{9999,9901}     	99000099
{99979,99681}   	9966006699
{999999,999001} 	999000000999
{9997647,9998017}	99956644665999

Paper & Pencil

find two 3-digit factors, that when multiplied together, yield the highest 6-digit palindrome.

lowest possible 6 digit palindrome starting with 9 is 900009
floor(sqrt(900009)) = 948
one factor must be an odd multiple of 11
floor(948 / 11) = 86
it must not be even or a multiple of 5, so use 83
83 * 11 = 913 <- this is the lowest possible first factor
the last digit of the second factor must multiply with the last digit of the first factor to get 9
the highest suitable second factor (for 913) is 993
913 x 993 = 906609, a palindrome, now check suitable higher first factors
913 + 22 = 935, an unsuitable multiple of 5, so skip it and use 913 + 44 = 957
957 x 997 = ‭954129‬, not a palindrome, so continue (just subtract 9570)
957 x 987 = 944559‬‬, not a palindrome, so continue
957 x 977 = ‭934989‬, not a palindrome, so continue
957 x 967 = ‭925429‬, not a palindrome, so continue
957 x 957 = ‭915849‬, not a palindrome, so continue
957 x 947 = ‭906279‬, not a palindrome, and less than the best found so far, so stop and
continue to the next suitible first number, 957 + 22 = 979
979 x 991 = 970189‬‬, not a palindrome, so continue (just subtract 9790)
979 x 981 = 960399‬, not a palindrome, so continue
979 x 971 = 950609‬, not a palindrome, so continue
979 x 961 = 940819‬, not a palindrome, so continue
979 x 951 = 931029‬, not a palindrome, so continue
979 x 941 = 921239‬, not a palindrome, so continue
979 x 931 = 911449‬, not a palindrome, so continue
979 x 921 = 901659‬, not a palindrome, and less than the best found so far, so stop
done because 979 + 22 = 1001

Perl

Library: ntheory

use strict;
use warnings;
use feature 'say';
use ntheory 'divisors';

for my $l (2..7) {
    LOOP:
    for my $p (reverse map { $_ . reverse $_ } 10**($l-1) .. 10**$l - 1)  {
        my @f = reverse grep { length == $l } divisors $p;
        next unless @f >= 2 and $p == $f[0] * $f[1];
        say "Largest palindromic product of two @{[$l]}-digit integers: $f[1] × $f[0] = $p" and last LOOP;
    }
}

Output:

Largest palindromic product of two 2-digit integers: 91 × 99 = 9009
Largest palindromic product of two 3-digit integers: 913 × 993 = 906609
Largest palindromic product of two 4-digit integers: 9901 × 9999 = 99000099
Largest palindromic product of two 5-digit integers: 99681 × 99979 = 9966006699
Largest palindromic product of two 6-digit integers: 999001 × 999999 = 999000000999
Largest palindromic product of two 7-digit integers: 9997647 × 9998017 = 99956644665999

Phix

Library: Phix/online

Translated from python by Lucy_Hedgehog as found on page 5 of the project euler discussion page (dated 25 Sep 2011), and further optimised as per the C# comments (on this very rosettacode page). You can run this online here.

-- demo\rosetta\Largest_palindrome_product.exw
with javascript_semantics
requires("1.0.1") -- (mpz_fdiv_qr(), mpz_si_sub() added to mpfr.js, mpz_mod_ui(), mpz_fdiv_q_ui(), mpz_fdiv_r(), mpz_fdiv_ui() fixed)
include mpfr.e

function ispalindrome(mpz x)
    string s = mpz_get_str(x)
    return s == reverse(s)
end function

function inverse(mpz x, integer m)
-- Compute the modular inverse of x modulo power(10,m).
-- Return -1 if the inverse does not exist.
-- This function uses Hensel lifting.
    integer a = {-1, 1, -1, 7, -1, -1, -1, 3, -1, 9}[mpz_fdiv_ui(x,10)+1]
    if a!=-1 then
        mpz ax = mpz_init()
        while true do
            mpz_mul_si(ax,x,a)
            {} = mpz_mod_ui(ax,ax,m)
            if mpz_cmp_si(ax,1)==0 then exit end if
            mpz_si_sub(ax,2,ax)
            mpz_mul_si(ax,ax,a)
            a = mpz_fdiv_q_ui(ax,ax,m)
        end while
    end if
    return a
end function

function pal2(integer n)
    assert(n>1)

    mpz {best,factor,y,r} = mpz_inits(4)
    if even(n) then
        // (as per the C# comments)
        mpz_ui_pow_ui(factor,10,n/2)
        mpz_sub_si(factor,factor,1)
        mpz_ui_pow_ui(best,10,n/2*3)
        mpz_mul(best,best,factor)
        mpz_add(best,best,factor)
        assert(ispalindrome(best))
        mpz_ui_pow_ui(factor,10,n)
        mpz_sub_si(factor,factor,1)
        assert(ispalindrome(factor))
    else
        // Get a lower bound:
        integer k = floor(n/2)
        mpz {maxf,maxf11,minf,x,t,maxy,p} = mpz_inits(7)
        while true do
            mpz_ui_pow_ui(maxf,10,n)
            mpz_sub_si(maxf,maxf,1)
            mpz_sub_si(maxf11,maxf,11)
            {} = mpz_fdiv_q_ui(maxf11,maxf11,22)
            mpz_mul_si(maxf11,maxf11,22)
            mpz_add_si(maxf11,maxf11,11)
            mpz_ui_pow_ui(minf,10,n-k)
            mpz_sub(minf,maxf,minf)
            mpz_add_si(minf,minf,2)
            mpz_mul(best,minf,minf)
            mpz_set_si(factor,0)
            // This palindrome starts with k 9's.
            // Hence the largest palindrom must also start with k 9's and
            // therefore end with k 9's.
            // Thus, if p = x * y is the solution then
            // x * y + 1 is divisible by m.
            integer m = power(10,k) -- (should not exceed 1e8)
            mpz_set(x,maxf11)
            while mpz_cmp_si(x,1)>=0 do
                mpz_mul(t,x,maxf)
                if mpz_cmp(t,best)=-1 then exit end if
                integer ry = inverse(x, m)
                if ry!=-1 then
                    mpz_add_si(maxy,maxf,1-ry)
                    mpz_mul(p,maxy,x)
                    while mpz_cmp(p,best)>0 do
                        if ispalindrome(p) then
                            mpz_set(best,p)
                            mpz_set(factor,x)
                        end if
                        mpz_mul_si(t,x,m)
                        mpz_sub(p,p,t)
                    end while
                end if
                mpz_sub_si(x,x,22)
            end while
            if mpz_cmp_si(factor,0)!=0 then exit end if
            k -= 1
        end while
    end if
    mpz_fdiv_qr(y,r,best,factor)
    assert(mpz_cmp_si(r,0)=0)
    return {best, factor, y}
end function

constant fmt = "Largest palindromic product of two %d-digit integers: %s = %s x %s (%s)\n"
for n=2 to 12 do
    atom t1 = time()
    mpz {p,x,y} = pal2(n)
    string sp = mpz_get_str(p),
           sx = mpz_get_str(x),
           sy = mpz_get_str(y),
            e = elapsed(time()-t1)
    printf(1,fmt,{n,sp,sx,sy,e})
end for

Output:

Largest palindromic product of two 2-digit integers: 9009 = 99 x 91 (0s)
Largest palindromic product of two 3-digit integers: 906609 = 913 x 993 (0s)
Largest palindromic product of two 4-digit integers: 99000099 = 9999 x 9901 (0s)
Largest palindromic product of two 5-digit integers: 9966006699 = 99979 x 99681 (0s)
Largest palindromic product of two 6-digit integers: 999000000999 = 999999 x 999001 (0s)
Largest palindromic product of two 7-digit integers: 99956644665999 = 9997647 x 9998017 (0.0s)
Largest palindromic product of two 8-digit integers: 9999000000009999 = 99999999 x 99990001 (0s)
Largest palindromic product of two 9-digit integers: 999900665566009999 = 999920317 x 999980347 (0.8s)
Largest palindromic product of two 10-digit integers: 99999000000000099999 = 9999999999 x 9999900001 (0s)
Largest palindromic product of two 11-digit integers: 9999994020000204999999 = 99999996349 x 99999943851 (0.1s)
Largest palindromic product of two 12-digit integers: 999999000000000000999999 = 999999999999 x 999999000001 (0s)

After that it starts to struggle a bit:

Largest palindromic product of two 13-digit integers: 99999963342000024336999999 = 9999996340851 x 9999999993349 (40.4s)
Largest palindromic product of two 14-digit integers: 9999999000000000000009999999 = 99999999999999 x 99999990000001 (0s)
Largest palindromic product of two 15-digit integers: 999999974180040040081479999999 = 999999998341069 x 999999975838971 (1 minute and 12s)
Largest palindromic product of two 16-digit integers: 99999999000000000000000099999999 = 9999999999999999 x 9999999900000001 (0s)

Python

Original author credit to user peijunz at Leetcode.

""" taken from https://leetcode.com/problems/largest-palindrome-product/discuss/150954/Fast-algorithm-by-constrains-on-tail-digits """

T=[set([(0, 0)])]

def double(it):
    for a, b in it:
        yield a, b
        yield b, a

def tails(n):
    '''Construct pair of n-digit numbers that their product ends with 99...9 pattern'''
    if len(T)<=n:
        l = set()
        for i in range(10):
            for j in range(i, 10):
                I = i*10**(n-1)
                J = j*10**(n-1)
                it = tails(n-1)
                if I!=J: it = double(it)
                for t1, t2 in it:
                    if ((I+t1)*(J+t2)+1)%10**n == 0:
                        l.add((I+t1, J+t2))
        T.append(l)
    return T[n]

def largestPalindrome(n):
    """ find largest palindrome that is a product of two n-digit numbers """
    m, tail = 0, n // 2
    head = n - tail
    up = 10**head
    for L in range(1, 9*10**(head-1)+1):
        # Consider small shell (up-L)^2 < (up-i)*(up-j) <= (up-L)^2, 1<=i<=L<=j
        m = 0
        sol = None
        for i in range(1, L + 1):
            lo = max(i, int(up - (up - L + 1)**2 / (up - i)) + 1)
            hi = int(up - (up - L)**2 / (up - i))
            for j in range(lo, hi + 1):
                I = (up-i) * 10**tail
                J = (up-j) * 10**tail
                it = tails(tail)
                if I!=J: it = double(it)
                    for t1, t2 in it:
                        val = (I + t1)*(J + t2)
                        s = str(val)
                        if s == s[::-1] and val>m:
                            sol = (I + t1, J + t2)
                            m = val

        if m:
            print("{:2d}\t{:4d}".format(n, m % 1337), sol, sol[0] * sol[1])
            return m % 1337
    return 0

if __name__ == "__main__":
    for k in range(1, 14):
        largestPalindrome(k)

Output:

 1	   9 (9, 1) 9
 2	 987 (91, 99) 9009
 3	 123 (993, 913) 906609
 4	 597 (9901, 9999) 99000099
 5	 677 (99979, 99681) 9966006699
 6	1218 (999001, 999999) 999000000999
 7	 877 (9998017, 9997647) 99956644665999
 8	 475 (99990001, 99999999) 9999000000009999
 9	1226 (999980347, 999920317) 999900665566009999
10	 875 (9999986701, 9999996699) 99999834000043899999
11	 108 (99999943851, 99999996349) 9999994020000204999999
12	 378 (999999000001, 999999999999) 999999000000000000999999
13	1097 (9999999993349, 9999996340851) 99999963342000024336999999

Raku

use Inline::Perl5;
my $p5 = Inline::Perl5.new();
$p5.use: 'ntheory';
my &divisors = $p5.run('sub { ntheory::divisors $_[0] }');

.say for (2..12).map: {.&lpp};

multi lpp ($oom where {!($_ +& 1)}) { # even number of multiplicand digits
    my $f = +(9 x $oom);
    my $o = $oom / 2;
    my $pal = +(9 x $o ~ 0 x $oom ~ 9 x $o);
    sprintf "Largest palindromic product of two %2d-digit integers: %d × %d = %d", $oom, $pal div $f, $f, $pal
}

multi lpp ($oom where {$_ +& 1}) { # odd number of multiplicand digits
    my $p;
    (+(1 ~ (0 x ($oom - 1))) .. +(9 ~ (9 x ($oom - 1)))).reverse.map({ +($_ ~ .flip) }).map: -> $pal {
        for my @factors = divisors("$pal")».Int.grep({ .chars == $oom }).sort( -* ) {
            next unless $pal div $_ ∈ @factors;
            $p = sprintf("Largest palindromic product of two %2d-digit integers: %d × %d = %d", $oom, $pal div $_, $_, $pal);
            last;
        }
        last if $p;
    }
    $p
}

Largest palindromic product of two  2-digit integers: 91 × 99 = 9009
Largest palindromic product of two  3-digit integers: 913 × 993 = 906609
Largest palindromic product of two  4-digit integers: 9901 × 9999 = 99000099
Largest palindromic product of two  5-digit integers: 99681 × 99979 = 9966006699
Largest palindromic product of two  6-digit integers: 999001 × 999999 = 999000000999
Largest palindromic product of two  7-digit integers: 9997647 × 9998017 = 99956644665999
Largest palindromic product of two  8-digit integers: 99990001 × 99999999 = 9999000000009999
Largest palindromic product of two  9-digit integers: 999920317 × 999980347 = 999900665566009999
Largest palindromic product of two 10-digit integers: 9999900001 × 9999999999 = 99999000000000099999
Largest palindromic product of two 11-digit integers: 99999943851 × 99999996349 = 9999994020000204999999
Largest palindromic product of two 12-digit integers: 999999000001 × 999999999999 = 999999000000000000999999

Ring

? "working..."

prod = 1
bestProd = 0
// maximum 3 digit number
max = 999
// both factors must be >100 for a 6 digit product 
limitStart = 101
// one factor must be divisible by 11
limitEnd = 11 * floor(max / 11)
second = limitStart
iters = 0

// loop from hi to low to find the best result in the fewest steps
for n = limitEnd to limitStart step -11
    // with n falling, the lower limit of m can rise with
    // the best-found-so-far second number. Doing this
    // lowers the iteration count by a lot.
    for m = max to second step -2
        prod = n * m
        if isPal(prod)
            iters++
            // exit when the product stops increasing
            if bestProd > prod
                exit
            ok
            // maintain the best-found-so-far result
            first = n
            second = m
            bestProd = prod
        ok
    next
next

put "The largest palindrome is: "
? "" + bestProd + " = " + first + " * " + second
? "Found in " + iters + " iterations"
put "done..."

func isPal n
    x = string(n)
    l = len(x) + 1
    i = 0
    while i < l
        if x[i++] != x[l--]
            return false
        ok
    end
    return true

Output:

working...
The largest palindrome is: 906609 = 913 * 993
Found in 6 iterations
done...

Sidef

func largest_palindrome_product (n) {

    for k in ((10**n - 1) `downto` 10**(n-1)) {
        var t = Num("#{k}#{Str(k).flip}")

        t.divisors.each {|d|
            if ((d.len == n) && ((t/d).len == n)) {
                return (d, t/d)
            }
        }
    }
}

for n in (2..9) {
    var (a,b) = largest_palindrome_product(n)
    say "Largest palindromic product of two #{n}-digit integers: #{a} * #{b} = #{a*b}"
}

Output:

Largest palindromic product of two 2-digit integers: 91 * 99 = 9009
Largest palindromic product of two 3-digit integers: 913 * 993 = 906609
Largest palindromic product of two 4-digit integers: 9901 * 9999 = 99000099
Largest palindromic product of two 5-digit integers: 99681 * 99979 = 9966006699
Largest palindromic product of two 6-digit integers: 999001 * 999999 = 999000000999
Largest palindromic product of two 7-digit integers: 9997647 * 9998017 = 99956644665999
Largest palindromic product of two 8-digit integers: 99990001 * 99999999 = 9999000000009999
Largest palindromic product of two 9-digit integers: 999920317 * 999980347 = 999900665566009999

Wren

The approach here is to manufacture palindromic numbers of length 2n in decreasing order and then see if they're products of two n-digit numbers.

var reverse = Fn.new { |n|
    var r = 0 
    while (n > 0) {
        r = n%10 + r*10
        n = (n/10).floor
    }
    return r
}

var pow = 10
for (n in 2..7) {
    var low = pow * 9
    pow = pow * 10
    var high = pow - 1
    System.write("Largest palindromic product of two %(n)-digit integers: ")
    var nextN = false
    for (i in high..low) {
        var j = reverse.call(i)
        var p = i * pow + j
        // k can't be even nor end in 5 to produce a product ending in 9
        var k = high
        while (k > low) {
            if (k % 10 != 5) {
                var l = p / k
                if (l > high) break
                if (p % k == 0) {
                    System.print("%(k) x %(l) = %(p)")
                    nextN = true
                    break
                }
            }
            k = k - 2
        }
        if (nextN) break
    }
}

Output:

Largest palindromic product of two 2-digit integers: 99 x 91 = 9009
Largest palindromic product of two 3-digit integers: 993 x 913 = 906609
Largest palindromic product of two 4-digit integers: 9999 x 9901 = 99000099
Largest palindromic product of two 5-digit integers: 99979 x 99681 = 9966006699
Largest palindromic product of two 6-digit integers: 999999 x 999001 = 999000000999
Largest palindromic product of two 7-digit integers: 9998017 x 9997647 = 99956644665999

XPL0

func Rev(A);    \Reverse digits
int A, B;
[B:= 0;
repeat  A:= A/10;
        B:= B*10 + rem(0);
until   A = 0;
return B;
];

int Max, M, N, Prod;
[Max:= 0;
for M:= 100 to 999 do
    for N:= M to 999 do
        [Prod:= M*N;
        if Prod/1000 = Rev(rem(0)) then
            if Prod > Max then Max:= Prod;
        ];
IntOut(0, Max);
]

Output:

906609