4-rings or 4-squares puzzle
- 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
REXX
<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). */
- =0 /*number of solutions found (so far). */
do a=LO to HI
do b=LO to HI
if unique then if b==a then iterate
do c=LO to HI
if unique then do; if c==a then iterate
if c==b then iterate
end
do d=LO to HI
if unique then do; if d==a then iterate
if d==b then iterate
if d==c then iterate
end
do e=LO to HI
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 to HI
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 to HI
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.
zkl
<lang zkl> // unique: No repeated numbers in solution fcn fourSquaresPuzzle(lo=1,hi=7,unique=True){ //-->list of solutions
notUnic:=fcn(a,b,c,etc){ (abc:=vm.arglist).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]){
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){
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())) }
println(solutions.len(),msg," solutions found.");
}
fourSquaresPuzzle() : show(_," unique (1-7)"); println(); fourSquaresPuzzle(3,9) : show(_," unique (3-9)"); println(); println(fourSquaresPuzzle(0,9,False).len(), // 10^7 possibilities
" non-unique (0-9) solutions found.");</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 (1-7) 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 (3-9) solutions found. 2860 non-unique (0-9) solutions found.