4-rings or 4-squares puzzle: Difference between revisions
| Line 680: | Line 680: | ||
2860 non-unique solutions found using 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. |
2860 non-unique solutions found using 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. |
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</pre> |
</pre> |
||
=={{header|Phix}}== |
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<lang Phix>integer solutions |
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procedure check(sequence set, integer show) |
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integer {a,b,c,d,e,f,g} = set |
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integer ab = a+b |
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if ab=b+d+c and ab=d+e+f and ab=f+g then |
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solutions += 1 |
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if show then |
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?set |
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end if |
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end if |
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end procedure |
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procedure foursquares(integer lo, integer hi, bool uniq, bool show) |
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sequence set = repeat(lo,7) |
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solutions = 0 |
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if uniq then |
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for i=1 to 7 do |
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set[i] = lo+i-1 |
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end for |
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for i=1 to factorial(7) do |
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check(permute(i,set),show) |
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end for |
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else |
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integer done = 0 |
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while not done do |
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check(set,show) |
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for i=1 to 7 do |
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set[i]+=1 |
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if set[i]<=hi then exit end if |
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if i=7 then |
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done = 1 |
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exit |
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end if |
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set[i] = lo |
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end for |
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end while |
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end if |
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printf(1,"%d solutions\n",solutions) |
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end procedure |
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foursquares(1,7,uniq:=True,show:=True) |
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foursquares(3,9,uniq:=True,show:=True) |
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foursquares(0,9,uniq:=False,show:=False)</lang> |
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{{out}} |
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<pre> |
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{6,4,5,1,2,7,3} |
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{3,7,2,1,5,4,6} |
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{6,4,1,5,2,3,7} |
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{4,7,1,3,2,6,5} |
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{7,3,2,5,1,4,6} |
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{5,6,2,3,1,7,4} |
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{4,5,3,1,6,2,7} |
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{7,2,6,1,3,5,4} |
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8 solutions |
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{7,8,3,4,5,6,9} |
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{8,7,3,5,4,6,9} |
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{9,6,4,5,3,7,8} |
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{9,6,5,4,3,8,7} |
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4 solutions |
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2860 solutions |
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</pre> |
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=={{header|PL/SQL}}== |
=={{header|PL/SQL}}== |
||
{{works with|Oracle}} |
{{works with|Oracle}} |
||
Revision as of 04:33, 3 February 2017
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Replace a, b, c, d, e, f, and
g with the decimal
digits LOW ───► HIGH
such that the sum of the letters inside of each of the four large squares add up to
the same sum.
╔══════════════╗ ╔══════════════╗
║ ║ ║ ║
║ a ║ ║ e ║
║ ║ ║ ║
║ ┌───╫──────╫───┐ ┌───╫─────────┐
║ │ ║ ║ │ │ ║ │
║ │ b ║ ║ d │ │ f ║ │
║ │ ║ ║ │ │ ║ │
║ │ ║ ║ │ │ ║ │
╚══════════╪═══╝ ╚═══╪══════╪═══╝ │
│ c │ │ g │
│ │ │ │
│ │ │ │
└──────────────┘ └─────────────┘
Show all output here.
- Show all solutions for each letter being unique with
LOW=1 HIGH=7
- Show all solutions for each letter being unique with
LOW=3 HIGH=9
- Show only the number of solutions when each letter can be non-unique
LOW=0 HIGH=9
- Related task
ALGOL 68
As with the REXX solution, we use explicit loops to generate the permutations. <lang algol68>BEGIN
# solve the 4 rings or 4 squares puzzle #
# we need to find solutions to the equations: a + b = b + c + d = d + e + f = f + g #
# where a, b, c, d, e, f, g in lo : hi ( not necessarily unique ) #
# depending on show, the solutions will be printed or not #
PROC four rings = ( INT lo, hi, BOOL unique, show )VOID:
BEGIN
INT solutions := 0;
BOOL allow duplicates = NOT unique;
# calculate field width for printinhg solutions #
INT width := -1;
INT max := ABS IF ABS lo > ABS hi THEN lo ELSE hi FI;
WHILE max > 0 DO
width -:= 1;
max OVERAB 10
OD;
# find solutions #
FOR a FROM lo TO hi DO
FOR b FROM lo TO hi DO
IF allow duplicates OR a /= b THEN
INT t = a + b;
FOR c FROM lo TO hi DO
IF allow duplicates OR ( a /= c AND b /= c ) THEN
FOR d FROM lo TO hi DO
IF allow duplicates OR ( a /= d AND b /= d AND c /= d )
THEN
IF b + c + d = t THEN
FOR e FROM lo TO hi DO
IF allow duplicates
OR ( a /= e AND b /= e AND c /= e AND d /= e )
THEN
FOR f FROM lo TO hi DO
IF allow duplicates
OR ( a /= f AND b /= f AND c /= f AND d /= f AND e /= f )
THEN
IF d + e + f = t THEN
FOR g FROM lo TO hi DO
IF allow duplicates
OR ( a /= g AND b /= g AND c /= g AND d /= g AND e /= g AND f /= g )
THEN
IF f + g = t THEN
solutions +:= 1;
IF show THEN
print( ( whole( a, width ), whole( b, width )
, whole( c, width ), whole( d, width )
, whole( e, width ), whole( f, width )
, whole( g, width ), newline
)
)
FI
FI
FI
OD # g #
FI
FI
OD # f #
FI
OD # e #
FI
FI
OD # d #
FI
OD # c #
FI
OD # b #
OD # a # ;
print( ( whole( solutions, 0 )
, IF unique THEN " unique" ELSE " non-unique" FI
, " solutions in "
, whole( lo, 0 )
, " to "
, whole( hi, 0 )
, newline
, newline
)
)
END # four rings # ;
# find the solutions as required for the task # four rings( 1, 7, TRUE, TRUE ); four rings( 3, 9, TRUE, TRUE ); four rings( 0, 9, FALSE, FALSE )
END</lang>
- Output:
3 7 2 1 5 4 6 4 5 3 1 6 2 7 4 7 1 3 2 6 5 5 6 2 3 1 7 4 6 4 1 5 2 3 7 6 4 5 1 2 7 3 7 2 6 1 3 5 4 7 3 2 5 1 4 6 8 unique solutions in 1 to 7 7 8 3 4 5 6 9 8 7 3 5 4 6 9 9 6 4 5 3 7 8 9 6 5 4 3 8 7 4 unique solutions in 3 to 9 2860 non-unique solutions in 0 to 9
C
<lang C>
- include <stdio.h>
- define TRUE 1
- define FALSE 0
int a,b,c,d,e,f,g; int lo,hi,unique,show; int solutions;
void bf() {
for (f = lo;f <= hi; f++)
if ((!unique) ||
((f != a) && (f != c) && (f != d) && (f != g) && (f != e)))
{
b = e + f - c;
if ((b >= lo) && (b <= hi) &&
((!unique) || ((b != a) && (b != c) &&
(b != d) && (b != g) && (b != e) && (b != f))))
{
solutions++;
if (show)
printf("%d %d %d %d %d %d %d\n",a,b,c,d,e,f,g);
}
}
}
void
ge()
{
for (e = lo;e <= hi; e++)
if ((!unique) || ((e != a) && (e != c) && (e != d)))
{
g = d + e;
if ((g >= lo) && (g <= hi) &&
((!unique) || ((g != a) && (g != c) &&
(g != d) && (g != e))))
bf();
}
}
void acd() {
for (c = lo;c <= hi; c++)
for (d = lo;d <= hi; d++)
if ((!unique) || (c != d))
{
a = c + d;
if ((a >= lo) && (a <= hi) &&
((!unique) || ((c != 0) && (d != 0))))
ge();
}
}
void
foursquares(int plo,int phi, int punique,int pshow)
{
lo = plo; hi = phi; unique = punique; show = pshow; solutions = 0;
printf("\n");
acd();
if (unique)
printf("\n%d unique solutions in %d to %d\n",solutions,lo,hi);
else
printf("\n%d non-unique solutions in %d to %d\n",solutions,lo,hi);
}
main() {
foursquares(1,7,TRUE,TRUE); foursquares(3,9,TRUE,TRUE); foursquares(0,9,FALSE,FALSE);
} </lang> Output
4 7 1 3 2 6 5 6 4 1 5 2 3 7 3 7 2 1 5 4 6 5 6 2 3 1 7 4 7 3 2 5 1 4 6 4 5 3 1 6 2 7 6 4 5 1 2 7 3 7 2 6 1 3 5 4 8 unique solutions in 1 to 7 7 8 3 4 5 6 9 8 7 3 5 4 6 9 9 6 4 5 3 7 8 9 6 5 4 3 8 7 4 unique solutions in 3 to 9 2860 non-unique solutions in 0 to 9
Common Lisp
<lang lisp> (defpackage four-rings
(:use common-lisp) (:export display-solutions))
(in-package four-rings)
(defun correct-answer-p (a b c d e f g)
(let ((v (+ a b)))
(and (equal v (+ b c d))
(equal v (+ d e f))
(equal v (+ f g)))))
(defun combinations-if (func len unique min max)
(let ((results nil))
(labels ((inner (cur)
(if (eql (length cur) len)
(when (apply func (reverse cur))
(push cur results))
(dotimes (i (- max min))
(when (or (not unique)
(not (member (+ i min) cur)))
(inner (append (list (+ i min)) cur)))))))
(inner nil))
results))
(defun four-rings-solutions (low high unique)
(combinations-if #'correct-answer-p 7 unique low (1+ high)))
(defun display-solutions ()
(let ((letters '((a b c d e f g))))
(format t "Low 1, High 7, unique letters: ~%~{~{~3A~}~%~}~%"
(append letters (four-rings-solutions 1 7 t)))
(format t "Low 3, High 9, unique letters: ~%~{~{~3A~}~%~}~%"
(append letters (four-rings-solutions 3 9 t)))
(format t "Number of solutions for Low 0, High 9 non-unique:~%~A~%"
(length (four-rings-solutions 0 9 nil)))))
</lang> Output:
CL-USER> (four-rings:display-solutions) Low 1, High 7, unique letters: A B C D E F G 6 4 1 5 2 3 7 4 5 3 1 6 2 7 3 7 2 1 5 4 6 7 3 2 5 1 4 6 4 7 1 3 2 6 5 5 6 2 3 1 7 4 7 2 6 1 3 5 4 6 4 5 1 2 7 3 Low 3, High 9, unique letters: A B C D E F G 7 8 3 4 5 6 9 8 7 3 5 4 6 9 9 6 4 5 3 7 8 9 6 5 4 3 8 7 Number of solutions for Low 0, High 9 non-unique: 2860 NIL
F#
<lang fsharp> (* A simple function to generate the sequence
Nigel Galloway: January 31st., 2017 *)
type G = {d:int;x:int;b:int;f:int} let N n g =
{(max (n-g) n) .. (min (g-n) g)} |> Seq.collect(fun d->{(max (d+n+n) (n+n))..(min (g+g) (d+g+g))} |> Seq.collect(fun x ->
seq{for a in n .. g do for b in n .. g do if (a+b) = x then for c in n .. g do if (b+c+d) = x then yield b} |> Seq.collect(fun b ->
seq{for f in n .. g do for G in n .. g do if (f+G) = x then for e in n .. g do if (f+e+d) = x then yield f} |> Seq.map(fun f -> {d=d;x=x;b=b;f=f}))))
</lang> Then: <lang fsharp> printfn "%d" (Seq.length (N 0 9)) </lang>
- Output:
2860
<lang fsharp> (* A simple function to generate the sequence with unique values
Nigel Galloway: January 31st., 2017 *)
type G = {d:int;x:int;b:int;f:int} let N n g =
{(max (n-g) n) .. (min (g-n) g)} |> Seq.filter(fun d -> d <> 0) |> Seq.collect(fun d->{(max (d+n+n) (n+n)) .. (min (g+g) (d+g+g))} |> Seq.collect(fun x ->
seq{for a in n .. g do if a <> d then for b in n .. g do if (a+b) = x && b <> a && b <> d then for c in n .. g do if (b+c+d) = x && c <> d && c <> a && c <> b then yield b} |> Seq.collect(fun b ->
seq{for f in n .. g do if f <> d && f <> b && f <> (x-b) && f <> (x-d-b) then for G in n .. g do if (f+G) = x && G <> d && G <> b && G <> f && G <> (x-b) && G <> (x-d-b) then for e in n .. g do if (f+e+d) = x && e <> d && e <> b && e <> f && e <> G && e <> (x-b) && e <> (x-d-b) then yield f} |> Seq.map(fun f -> {d=d;x=x;b=b;f=f}))))
</lang> Then: <lang fsharp> for n in N 1 7 do printfn "%d,%d,%d,%d,%d,%d,%d" (n.x-n.b) n.b (n.x-n.d-n.b) n.d (n.x-n.d-n.f) n.f (n.x-n.f) </lang>
- Output:
4,5,3,1,6,2,7 7,2,6,1,3,5,4 3,7,2,1,5,4,6 6,4,5,1,2,7,3 4,7,1,3,2,6,5 5,6,2,3,1,7,4 6,4,1,5,2,3,7 7,3,2,5,1,4,6
and: <lang fsharp> for n in N 3 9 do printfn "%d,%d,%d,%d,%d,%d,%d" (n.x-n.b) n.b (n.x-n.d-n.b) n.d (n.x-n.d-n.f) n.f (n.x-n.f) </lang>
- Output:
7,8,3,4,5,6,9 9,6,5,4,3,8,7 8,7,3,5,4,6,9 9,6,4,5,3,7,8
Go
<lang go>package main
import "fmt"
func main(){ n, c := getCombs(1,7,true) fmt.Printf("%d unique solutions in 1 to 7\n",n) fmt.Println(c) n, c = getCombs(3,9,true) fmt.Printf("%d unique solutions in 3 to 9\n",n) fmt.Println(c) n, _ = getCombs(0,9,false) fmt.Printf("%d non-unique solutions in 0 to 9\n",n) }
func getCombs(low,high int,unique bool) (num int,validCombs [][]int){ for a := low; a <= high; a++ { for b := low; b <= high; b++ { for c := low; c <= high; c++ { for d := low; d <= high; d++ { for e := low; e <= high; e++ { for f := low; f <= high; f++ { for g := low; g <= high; g++ { if validComb(a,b,c,d,e,f,g) { if unique{ if isUnique(a,b,c,d,e,f,g) { num++ validCombs = append(validCombs,[]int{a,b,c,d,e,f,g}) } }else{ num++ validCombs = append(validCombs,[]int{a,b,c,d,e,f,g}) } } } } } } } } } return } func isUnique(a,b,c,d,e,f,g int) (res bool) { data := make(map[int]int) data[a]++ data[b]++ data[c]++ data[d]++ data[e]++ data[f]++ data[g]++ if len(data) == 7 { return true }else { return false } } func validComb(a,b,c,d,e,f,g int) bool{ square1 := a + b square2 := b + c + d square3 := d + e + f square4 := f + g return square1 == square2 && square2 == square3 && square3 == square4 } </lang>
- Output:
8 unique solutions in 1 to 7 [[3 7 2 1 5 4 6] [4 5 3 1 6 2 7] [4 7 1 3 2 6 5] [5 6 2 3 1 7 4] [6 4 1 5 2 3 7] [6 4 5 1 2 7 3] [7 2 6 1 3 5 4] [7 3 2 5 1 4 6]] 4 unique solutions in 3 to 9 [[7 8 3 4 5 6 9] [8 7 3 5 4 6 9] [9 6 4 5 3 7 8] [9 6 5 4 3 8 7]] 2860 non-unique solutions in 0 to 9
Pascal
There are so few solutions of 7 consecutive numbers, so I used a modified version, to get all the expected solutions at once. <lang pascal>program square4; {$MODE DELPHI} {$R+,O+} const
LoDgt = 0; HiDgt = 9;
type
tchkset = set of LoDgt..HiDgt;
tSol = record
solMin : integer;
solDat : array[1..7] of integer;
end;
var
sum,a,b,c,d,e,f,g,cnt,uniqueCount : NativeInt; sol : array of tSol;
procedure SolOut; var
i,j,mn: NativeInt;
Begin
mn := 0;
repeat
writeln(mn:3,' ...',mn+6:3);
For i := Low(sol) to High(sol) do
with sol[i] do
IF solMin = mn then
Begin
For j := 1 to 7 do
write(solDat[j]:3);
writeln;
end;
writeln;
inc(mn);
until mn > HiDgt-6;
end;
function CheckUnique:Boolean; var
i,sum,mn: NativeInt; chkset : tchkset;
Begin
chkset:= [];
include(chkset,a);include(chkset,b);include(chkset,c);
include(chkset,d);include(chkset,e);include(chkset,f);
include(chkset,g);
sum := 0;
For i := LoDgt to HiDgt do
IF i in chkset then
inc(sum);
result := sum = 7;
IF result then
begin
inc(uniqueCount);
//find the lowest entry
mn:= LoDgt;
For i := LoDgt to HiDgt do
IF i in chkset then
Begin
mn := i;
BREAK;
end;
// are they consecutive
For i := mn+1 to mn+6 do
IF NOT(i in chkset) then
EXIT;
setlength(sol,Length(sol)+1);
with sol[high(sol)] do
Begin
solMin:= mn;
solDat[1]:= a;solDat[2]:= b;solDat[3]:= c;
solDat[4]:= d;solDat[5]:= e;solDat[6]:= f;
solDat[7]:= g;
end;
end;
end;
Begin
cnt := 0;
uniqueCount := 0;
For a:= LoDgt to HiDgt do
Begin
For b := LoDgt to HiDgt do
Begin
sum := a+b;
//a+b = b+c+d => a = c+d => d := a-c
For c := a-LoDgt downto LoDgt do
begin
d := a-c;
e := sum-d;
IF e>HiDgt then
e:= HiDgt;
For e := e downto LoDgt do
begin
f := sum-e-d;
IF f in [loDGt..Hidgt]then
Begin
g := sum-f;
IF g in [loDGt..Hidgt]then
Begin
inc(cnt);
CheckUnique;
end;
end;
end;
end;
end;
end;
SolOut;
writeln(' solution count for ',loDgt,' to ',HiDgt,' = ',cnt);
writeln('unique solution count for ',loDgt,' to ',HiDgt,' = ',uniqueCount);
end.</lang>
- Output:
0 ... 6
4 2 3 1 5 0 6
5 1 3 2 4 0 6
6 0 5 1 3 2 4
6 0 4 2 3 1 5
1 ... 7
3 7 2 1 5 4 6
4 5 3 1 6 2 7
4 7 1 3 2 6 5
5 6 2 3 1 7 4
6 4 5 1 2 7 3
6 4 1 5 2 3 7
7 2 6 1 3 5 4
7 3 2 5 1 4 6
2 ... 8
5 7 3 2 6 4 8
5 8 3 2 4 7 6
5 8 2 3 4 6 7
6 7 4 2 3 8 5
7 4 5 2 6 3 8
7 6 4 3 2 8 5
8 3 6 2 5 4 7
8 4 6 2 3 7 5
3 ... 9
7 8 3 4 5 6 9
8 7 3 5 4 6 9
9 6 5 4 3 8 7
9 6 4 5 3 7 8
solution count for 0 to 9 = 2860
unique solution count for 0 to 9 = 192
Perl 6
<lang perl6>sub four-squares ( @list, :$unique=1, :$show=1 ) {
my @solutions;
for $unique.&combos -> @c {
@solutions.push: @c if [==]
@c[0] + @c[1],
@c[1] + @c[2] + @c[3],
@c[3] + @c[4] + @c[5],
@c[5] + @c[6];
}
say +@solutions, ($unique ?? ' ' !! ' non-'), "unique solutions found using {join(', ', @list)}.\n";
my $f = "%{@list.max.chars}s";
say join "\n", (('a'..'g').fmt: $f), @solutions».fmt($f), "\n" if $show;
multi combos ( $ where so * ) { @list.combinations(7).map: |*.permutations }
multi combos ( $ where not * ) { [X] @list xx 7 }
}
- TASK
four-squares( [1..7] ); four-squares( [3..9] ); four-squares( [8, 9, 11, 12, 17, 18, 20, 21] ); four-squares( [0..9], :unique(0), :show(0) );</lang>
- Output:
8 unique solutions found using 1, 2, 3, 4, 5, 6, 7. a b c d e f g 3 7 2 1 5 4 6 4 5 3 1 6 2 7 4 7 1 3 2 6 5 5 6 2 3 1 7 4 6 4 1 5 2 3 7 6 4 5 1 2 7 3 7 2 6 1 3 5 4 7 3 2 5 1 4 6 4 unique solutions found using 3, 4, 5, 6, 7, 8, 9. a b c d e f g 7 8 3 4 5 6 9 8 7 3 5 4 6 9 9 6 4 5 3 7 8 9 6 5 4 3 8 7 8 unique solutions found using 8, 9, 11, 12, 17, 18, 20, 21. a b c d e f g 17 21 8 9 11 18 20 20 18 11 9 8 21 17 17 21 9 8 12 18 20 20 18 8 12 9 17 21 20 18 12 8 9 21 17 21 17 9 12 8 18 20 20 18 11 9 12 17 21 21 17 12 9 11 18 20 2860 non-unique solutions found using 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Phix
<lang Phix>integer solutions
procedure check(sequence set, integer show)
integer {a,b,c,d,e,f,g} = set
integer ab = a+b
if ab=b+d+c and ab=d+e+f and ab=f+g then
solutions += 1
if show then
?set
end if
end if
end procedure
procedure foursquares(integer lo, integer hi, bool uniq, bool show) sequence set = repeat(lo,7)
solutions = 0
if uniq then
for i=1 to 7 do
set[i] = lo+i-1
end for
for i=1 to factorial(7) do
check(permute(i,set),show)
end for
else
integer done = 0
while not done do
check(set,show)
for i=1 to 7 do
set[i]+=1
if set[i]<=hi then exit end if
if i=7 then
done = 1
exit
end if
set[i] = lo
end for
end while
end if
printf(1,"%d solutions\n",solutions)
end procedure foursquares(1,7,uniq:=True,show:=True) foursquares(3,9,uniq:=True,show:=True) foursquares(0,9,uniq:=False,show:=False)</lang>
- Output:
{6,4,5,1,2,7,3}
{3,7,2,1,5,4,6}
{6,4,1,5,2,3,7}
{4,7,1,3,2,6,5}
{7,3,2,5,1,4,6}
{5,6,2,3,1,7,4}
{4,5,3,1,6,2,7}
{7,2,6,1,3,5,4}
8 solutions
{7,8,3,4,5,6,9}
{8,7,3,5,4,6,9}
{9,6,4,5,3,7,8}
{9,6,5,4,3,8,7}
4 solutions
2860 solutions
PL/SQL
<lang plsql> create table allints (v number); create table results ( a number, b number, c number, d number, e number, f number, g number );
create or replace procedure foursquares(lo number,hi number,uniq boolean,show boolean) as
a number;
b number;
c number;
d number;
e number;
f number;
g number;
out_line varchar2(2000);
cursor results_cur is
select
a,
b,
c,
d,
e,
f,
g
from
results
order by
a,b,c,d,e,f,g;
results_rec results_cur%rowtype; solutions number; uorn varchar2(2000);
begin
solutions := 0;
delete from allints;
delete from results;
for i in lo..hi loop
insert into allints values (i);
end loop;
commit;
if uniq = TRUE then
insert into results
select
a.v a,
b.v b,
c.v c,
d.v d,
e.v e,
f.v f,
g.v g
from
allints a, allints b, allints c,allints d,
allints e, allints f, allints g
where
a.v not in (b.v,c.v,d.v,e.v,f.v,g.v) and
b.v not in (c.v,d.v,e.v,f.v,g.v) and
c.v not in (d.v,e.v,f.v,g.v) and
d.v not in (e.v,f.v,g.v) and
e.v not in (f.v,g.v) and
f.v not in (g.v) and
a.v = c.v + d.v and
g.v = d.v + e.v and
b.v = e.v + f.v - c.v
order by
a,b,c,d,e,f,g;
uorn := ' unique solutions in ';
else
insert into results
select
a.v a,
b.v b,
c.v c,
d.v d,
e.v e,
f.v f,
g.v g
from
allints a, allints b, allints c,allints d,
allints e, allints f, allints g
where
a.v = c.v + d.v and
g.v = d.v + e.v and
b.v = e.v + f.v - c.v
order by
a,b,c,d,e,f,g;
uorn := ' non-unique solutions in ';
end if;
commit;
open results_cur;
loop
fetch results_cur into results_rec;
exit when results_cur%notfound;
a := results_rec.a;
b := results_rec.b;
c := results_rec.c;
d := results_rec.d;
e := results_rec.e;
f := results_rec.f;
g := results_rec.g;
solutions := solutions + 1;
if show = TRUE then
out_line := to_char(a) || ' ';
out_line := out_line || ' ' || to_char(b) || ' ';
out_line := out_line || ' ' || to_char(c) || ' ';
out_line := out_line || ' ' || to_char(d) || ' ';
out_line := out_line || ' ' || to_char(e) || ' ';
out_line := out_line || ' ' || to_char(f) ||' ';
out_line := out_line || ' ' || to_char(g);
end if;
dbms_output.put_line(out_line);
end loop;
close results_cur;
out_line := to_char(solutions) || uorn;
out_line := out_line || to_char(lo) || ' to ' || to_char(hi);
dbms_output.put_line(out_line);
end; / </lang> Output
SQL> execute foursquares(1,7,TRUE,TRUE); 3 7 2 1 5 4 6 4 5 3 1 6 2 7 4 7 1 3 2 6 5 5 6 2 3 1 7 4 6 4 1 5 2 3 7 6 4 5 1 2 7 3 7 2 6 1 3 5 4 7 3 2 5 1 4 6 8 unique solutions in 1 to 7 PL/SQL procedure successfully completed. SQL> execute foursquares(3,9,TRUE,TRUE); 7 8 3 4 5 6 9 8 7 3 5 4 6 9 9 6 4 5 3 7 8 9 6 5 4 3 8 7 4 unique solutions in 3 to 9 PL/SQL procedure successfully completed. SQL> execute foursquares(0,9,FALSE,FALSE); 2860 non-unique solutions in 0 to 9 PL/SQL procedure successfully completed.
Python
<lang Python> import itertools
def all_equal(a,b,c,d,e,f,g):
return a+b == b+c+d and a+b == d+e+f and a+b == f+g
def foursquares(lo,hi,unique,show):
solutions = 0
if unique:
uorn = "unique"
citer = itertools.combinations(range(lo,hi+1),7)
else:
uorn = "non-unique"
citer = itertools.combinations_with_replacement(range(lo,hi+1),7)
for c in citer:
for p in set(itertools.permutations(c)):
if all_equal(*p):
solutions += 1
if show:
print str(p)[1:-1]
print str(solutions)+" "+uorn+" solutions in "+str(lo)+" to "+str(hi) print
</lang> Output
foursquares(1,7,True,True) 4, 5, 3, 1, 6, 2, 7 3, 7, 2, 1, 5, 4, 6 5, 6, 2, 3, 1, 7, 4 4, 7, 1, 3, 2, 6, 5 6, 4, 5, 1, 2, 7, 3 7, 3, 2, 5, 1, 4, 6 7, 2, 6, 1, 3, 5, 4 6, 4, 1, 5, 2, 3, 7 8 unique solutions in 1 to 7 foursquares(3,9,True,True) 7, 8, 3, 4, 5, 6, 9 9, 6, 4, 5, 3, 7, 8 8, 7, 3, 5, 4, 6, 9 9, 6, 5, 4, 3, 8, 7 4 unique solutions in 3 to 9 foursquares(0,9,False,False) 2860 non-unique solutions in 0 to 9
Faster solution without itertools <lang Python> def foursquares(lo,hi,unique,show):
def acd_iter():
"""
Iterates through all the possible valid values of
a, c, and d.
a = c + d
"""
for c in range(lo,hi+1):
for d in range(lo,hi+1):
if (not unique) or (c <> d):
a = c + d
if a >= lo and a <= hi:
if (not unique) or (c <> 0 and d <> 0):
yield (a,c,d)
def ge_iter():
"""
Iterates through all the possible valid values of
g and e.
g = d + e
"""
for e in range(lo,hi+1):
if (not unique) or (e not in (a,c,d)):
g = d + e
if g >= lo and g <= hi:
if (not unique) or (g not in (a,c,d,e)):
yield (g,e)
def bf_iter():
"""
Iterates through all the possible valid values of
b and f.
b = e + f - c
"""
for f in range(lo,hi+1):
if (not unique) or (f not in (a,c,d,g,e)):
b = e + f - c
if b >= lo and b <= hi:
if (not unique) or (b not in (a,c,d,g,e,f)):
yield (b,f)
solutions = 0
acd_itr = acd_iter()
for acd in acd_itr:
a,c,d = acd
ge_itr = ge_iter()
for ge in ge_itr:
g,e = ge
bf_itr = bf_iter()
for bf in bf_itr:
b,f = bf
solutions += 1
if show:
print str((a,b,c,d,e,f,g))[1:-1]
if unique:
uorn = "unique"
else:
uorn = "non-unique"
print str(solutions)+" "+uorn+" solutions in "+str(lo)+" to "+str(hi)
print
</lang> Output
foursquares(1,7,True,True) 4, 7, 1, 3, 2, 6, 5 6, 4, 1, 5, 2, 3, 7 3, 7, 2, 1, 5, 4, 6 5, 6, 2, 3, 1, 7, 4 7, 3, 2, 5, 1, 4, 6 4, 5, 3, 1, 6, 2, 7 6, 4, 5, 1, 2, 7, 3 7, 2, 6, 1, 3, 5, 4 8 unique solutions in 1 to 7 foursquares(3,9,True,True) 7, 8, 3, 4, 5, 6, 9 8, 7, 3, 5, 4, 6, 9 9, 6, 4, 5, 3, 7, 8 9, 6, 5, 4, 3, 8, 7 4 unique solutions in 3 to 9 foursquares(0,9,False,False) 2860 non-unique solutions in 0 to 9
REXX
fast version
This REXX version is faster than the more idiomatic version, but is longer (statement-wise) and
a bit easier to read (visualize).
<lang rexx>/*REXX pgm solves the 4-rings puzzle, where letters represent unique (or not) digits). */
arg LO HI unique show . /*the ARG statement capitalizes args.*/
if LO== | LO=="," then LO=1 /*Not specified? Then use the default.*/
if HI== | HI=="," then HI=7 /* " " " " " " */
if unique== | unique==',' | unique=='UNIQUE' then unique=1 /*unique letter solutions*/
else unique=0 /*non-unique " */
if show== | show==',' | show=='SHOW' then show=1 /*noshow letter solutions*/
else show=0 /* show " " */
w=max(3, length(LO), length(HI) ) /*maximum width of any number found. */ bar=copies('═', w) /*define a horizontal bar (for title). */ times=HI - LO + 1 /*calculate number of times to loop. */
- =0 /*number of solutions found (so far). */
do a=LO for times
do b=LO for times
if unique then if b==a then iterate
do c=LO for times
if unique then do; if c==a then iterate
if c==b then iterate
end
do d=LO for times
if unique then do; if d==a then iterate
if d==b then iterate
if d==c then iterate
end
do e=LO for times
if unique then do; if e==a then iterate
if e==b then iterate
if e==c then iterate
if e==d then iterate
end
do f=LO for times
if unique then do; if f==a then iterate
if f==b then iterate
if f==c then iterate
if f==d then iterate
if f==e then iterate
end
do g=LO for times
if unique then do; if g==a then iterate
if g==b then iterate
if g==c then iterate
if g==d then iterate
if g==e then iterate
if g==f then iterate
end
sum=a+b
if f+g\==sum then iterate
if b+c+d\==sum then iterate
if d+e+f\==sum then iterate
#=# + 1 /*bump the count of solutions.*/
if #==1 then call align 'a', 'b', 'c', 'd', 'e', 'f', 'g'
if #==1 then call align bar, bar, bar, bar, bar, bar, bar
call align a, b, c, d, e, f, g
end /*g*/
end /*f*/
end /*e*/
end /*d*/
end /*c*/
end /*b*/
end /*a*/
say
_= ' non-unique'
if unique then _= ' unique ' say # _ 'solutions found.' exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ align: parse arg a1,a2,a3,a4,a5,a6,a7
if show then say left(,9) center(a1,w) center(a2,w) center(a3,w) center(a4,w),
center(a5,w) center(a6,w) center(a7,w)
return</lang>
output when using the default inputs: 1 7
a b c d e f g
═══ ═══ ═══ ═══ ═══ ═══ ═══
3 7 2 1 5 4 6
4 5 3 1 6 2 7
4 7 1 3 2 6 5
5 6 2 3 1 7 4
6 4 1 5 2 3 7
6 4 5 1 2 7 3
7 2 6 1 3 5 4
7 3 2 5 1 4 6
8 unique solutions found.
output when using the input of: 3 9
a b c d e f g
═══ ═══ ═══ ═══ ═══ ═══ ═══
7 8 3 4 5 6 9
8 7 3 5 4 6 9
9 6 4 5 3 7 8
9 6 5 4 3 8 7
4 unique solutions found.
output when using the input of: 0 9 non-unique noshow
2860 non-unique solutions found.
idiomatic version
This REXX version is slower than the faster version (because of the multiple if clauses.
Note that the REXX language doesn't have short-circuits (when executing multiple clauses in if (and other) statements. <lang rexx>/*REXX pgm solves the 4-rings puzzle, where letters represent unique (or not) digits). */ arg LO HI unique show . /*the ARG statement capitalizes args.*/ if LO== | LO=="," then LO=1 /*Not specified? Then use the default.*/ if HI== | HI=="," then HI=7 /* " " " " " " */ if unique== | unique==',' | unique=='UNIQUE' then u=1 /*unique letter solutions*/
else u=0 /*non-unique " */
if show== | show==',' | show=='SHOW' then show=1 /*noshow letter solutions*/
else show=0 /* show " " */
w=max(3, length(LO), length(HI) ) /*maximum width of any number found. */ bar=copies('═', w) /*define a horizontal bar (for title). */ times=HI - LO + 1 /*calculate number of times to loop. */
- =0 /*number of solutions found (so far). */
do a=LO for times
do b=LO for times
if u then if b==a then iterate
do c=LO for times
if u then if c==a | c==b then iterate
do d=LO for times
if u then if d==a | d==b | d==c then iterate
do e=LO for times
if u then if e==a | e==b | e==c | e==d then iterate
do f=LO for times
if u then if f==a | f==b | f==c | f==d | f==e then iterate
do g=LO for times
if u then if g==a | g==b | g==c | g==d | g==e | g==f then iterate
sum=a+b
if f+g==sum & b+c+d==sum & d+e+f==sum then #=#+1 /*bump #.*/
else iterate /*a no-go*/
#=# + 1 /*bump the count of solutions.*/
if #==1 then call align 'a', 'b', 'c', 'd', 'e', 'f', 'g'
if #==1 then call align bar, bar, bar, bar, bar, bar, bar
call align a, b, c, d, e, f, g
end /*g*/ /*for the 1st time, show title*/
end /*f*/
end /*e*/
end /*d*/
end /*c*/
end /*b*/
end /*a*/
say
_= ' non-unique'
if u then _= ' unique ' say # _ 'solutions found.' exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ align: parse arg a1,a2,a3,a4,a5,a6,a7
if show then say left(,9) center(a1,w) center(a2,w) center(a3,w) center(a4,w),
center(a5,w) center(a6,w) center(a7,w)
return</lang>
output is identical to the faster REXX version.
Ruby
<lang ruby>def four_squares(low, high, unique=true, show=unique)
f = -> (a,b,c,d,e,f,g) {[a+b, b+c+d, d+e+f, f+g].uniq.size == 1}
if unique
uniq = "unique"
solutions = [*low..high].permutation(7).select{|ary| f.call(*ary)}
else
uniq = "non-unique"
solutions = [*low..high].repeated_permutation(7).select{|ary| f.call(*ary)}
end
if show
puts " " + [*"a".."g"].join(" ")
solutions.each{|ary| p ary}
end
puts "#{solutions.size} #{uniq} solutions in #{low} to #{high}"
puts
end
[[1,7], [3,9]].each do |low, high|
four_squares(low, high)
end four_squares(0, 9, false)</lang>
- Output:
a b c d e f g [3, 7, 2, 1, 5, 4, 6] [4, 5, 3, 1, 6, 2, 7] [4, 7, 1, 3, 2, 6, 5] [5, 6, 2, 3, 1, 7, 4] [6, 4, 1, 5, 2, 3, 7] [6, 4, 5, 1, 2, 7, 3] [7, 2, 6, 1, 3, 5, 4] [7, 3, 2, 5, 1, 4, 6] 8 unique solutions in 1 to 7 a b c d e f g [7, 8, 3, 4, 5, 6, 9] [8, 7, 3, 5, 4, 6, 9] [9, 6, 4, 5, 3, 7, 8] [9, 6, 5, 4, 3, 8, 7] 4 unique solutions in 3 to 9 2860 non-unique solutions in 0 to 9
zkl
<lang zkl> // unique: No repeated numbers in solution fcn fourSquaresPuzzle(lo=1,hi=7,unique=True){ //-->list of solutions
_assert_(0<=lo and hi<36);
notUnic:=fcn(a,b,c,etc){ abc:=vm.arglist; // use base 36, any repeated character?
abc.apply("toString",36).concat().unique().len()!=abc.len()
};
s:=List(); // solutions
foreach a,b,c in ([lo..hi],[lo..hi],[lo..hi]){ // chunk to reduce unique
if(unique and notUnic(a,b,c)) continue; // solution space. Slow VM
foreach d,e in ([lo..hi],[lo..hi]){ // -->for d { for e {} }
if(unique and notUnic(a,b,c,d,e)) continue;
foreach f,g in ([lo..hi],[lo..hi]){ if(unique and notUnic(a,b,c,d,e,f,g)) continue; sqr1,sqr2,sqr3,sqr4 := a+b,b+c+d,d+e+f,f+g; if((sqr1==sqr2==sqr3) and sqr1==sqr4) s.append(T(a,b,c,d,e,f,g)); }
} } s
}</lang> <lang zkl>fcn show(solutions,msg){
if(not solutions){ println("No solutions for",msg); return(); }
println(solutions.len(),msg," solutions found:");
w:=(1).max(solutions.pump(List,(0).max,"numDigits")); // max width of any number found
fmt:=" " + "%%%ds ".fmt(w)*7; // eg " %1s %1s %1s %1s %1s %1s %1s"
println(fmt.fmt(["a".."g"].walk().xplode()));
println("-"*((w+1)*7 + 1)); // calculate the width of horizontal bar
foreach s in (solutions){ println(fmt.fmt(s.xplode())) }
} fourSquaresPuzzle() : show(_," unique (1-7)"); println(); fourSquaresPuzzle(3,9) : show(_," unique (3-9)"); println(); fourSquaresPuzzle(5,12) : show(_," unique (5-12)"); println(); println(fourSquaresPuzzle(0,9,False).len(), // 10^7 possibilities
" non-unique (0-9) solutions found.");</lang>
- Output:
8 unique (1-7) solutions found: a b c d e f g --------------- 3 7 2 1 5 4 6 4 5 3 1 6 2 7 4 7 1 3 2 6 5 5 6 2 3 1 7 4 6 4 1 5 2 3 7 6 4 5 1 2 7 3 7 2 6 1 3 5 4 7 3 2 5 1 4 6 4 unique (3-9) solutions found: a b c d e f g --------------- 7 8 3 4 5 6 9 8 7 3 5 4 6 9 9 6 4 5 3 7 8 9 6 5 4 3 8 7 4 unique (5-12) solutions found: a b c d e f g ---------------------- 11 9 6 5 7 8 12 11 10 6 5 7 9 12 12 8 7 5 6 9 11 12 9 7 5 6 10 11 2860 non-unique (0-9) solutions found.