Arrange the digits 1-9 in a 3x3 square using each once only according to the following rules:

1. Exactly one prime is directly above a prime two less.



2. Each pair of opposite corners sums to the same square total and exactly two columns share another square total.



3. Exactly one prime is directly to the left of a non-prime two greater.



4. Directly above exactly one square digit is a digit four greater.



5. Directly to the right of exactly one cubic digit is a digit one greater.

By rule 1, a 7 has to be over a 5 or a 5 over a 3.

The pairs satisfying rule 3 could be 2,4 or 7,9.

By rule 4, either 8 is above 4 or 5 above 1.

The pairs satisfying rule 5 are 1,2 or 8,9.

The following program looks for such, as well as most of rule 2:

DECLARE SUB permute (a$)
CLS
s$ = "123456789"
FOR i = 1 TO 362880
 bad = 1
 ix = INSTR(s$, "7")
 IF ix < 7 THEN IF MID$(s$, ix + 3, 1) = \"5\" THEN bad = 0
 ix = INSTR(s$, \"5\")
 IF ix < 7 THEN IF MID$(s$, ix + 3, 1) = \"3\" THEN bad = 0
 IF bad THEN GOTO notThis

 bad = 1
 ix = INSTR(s$, \"8\")
 IF ix < 7 THEN IF MID$(s$, ix + 3, 1) = \"4\" THEN bad = 0
 ix = INSTR(s$, \"5\")
 IF ix < 7 THEN IF MID$(s$, ix + 3, 1) = \"1\" THEN bad = 0
 IF bad THEN GOTO notThis

 bad = 1
 ix = INSTR(s$, \"2\")
 IF ix MOD 3 THEN IF MID$(s$, ix + 1, 1) = \"4\" THEN bad = 0
 ix = INSTR(s$, \"7\")
 IF ix MOD 3 THEN IF MID$(s$, ix + 1, 1) = \"9\" THEN bad = 0
 IF bad THEN GOTO notThis

 bad = 1
 ix = INSTR(s$, \"1\")
 IF ix MOD 3 THEN IF MID$(s$, ix + 1, 1) = \"2\" THEN bad = 0
 ix = INSTR(s$, \"8\")
 IF ix MOD 3 THEN IF MID$(s$, ix + 1, 1) = \"9\" THEN bad = 0
 IF bad THEN GOTO notThis

sqCt = 0
 FOR col = 1 TO 3
   t(col) = 0
   FOR row = 1 TO 3
     t(col) = t(col) + VAL(MID$(s$, (row - 1) * 3 + col, 1))
   NEXT
   sqrt = INT(SQR(t(col)) + .5)
   IF sqrt * sqrt = t(col) THEN sqCt = sqCt + 1: sq(sqCt) = t(col)
 NEXT
 IF sqCt < 2 THEN GOTO notThis
 IF sq(1) <> sq(2) THEN GOTO notThis
 p1 = VAL(MID$(s$, 1, 1)) + VAL(MID$(s$, 9, 1))
 p2 = VAL(MID$(s$, 3, 1)) + VAL(MID$(s$, 7, 1))
 IF p1 <> p2 THEN GOTO notThis


 PRINT MID$(s$, 1, 3)
 PRINT MID$(s$, 4, 3)
 PRINT MID$(s$, 7, 3)
 PRINT
notThis:
 permute s$
NEXT

SUB permute (a$)
DEFINT A-Z
 x$ = ""
 FOR i = LEN(a$) TO 1 STEP -1
  l$ = x$
  x$ = MID$(a$, i, 1)
  IF x$ < l$ THEN EXIT FOR
 NEXT

 IF i = 0 THEN
  FOR j = 1 TO LEN(a$) \ 2
   x$ = MID$(a$, j, 1)
   MID$(a$, j, 1) = MID$(a$, LEN(a$) - j + 1, 1)
   MID$(a$, LEN(a$) - j + 1, 1) = x$
  NEXT
 ELSE
  FOR j = LEN(a$) TO i + 1 STEP -1
   IF MID$(a$, j, 1) > x$ THEN EXIT FOR
  NEXT
  MID$(a$, i, 1) = MID$(a$, j, 1)
  MID$(a$, j, 1) = x$
  FOR j = 1 TO (LEN(a$) - i) \ 2
   x$ = MID$(a$, i + j, 1)
   MID$(a$, i + j, 1) = MID$(a$, LEN(a$) - j + 1, 1)
   MID$(a$, LEN(a$) - j + 1, 1) = x$
  NEXT
 END IF
END SUB

and finds

789
524
163

897
635
241

The first does not satisfy the perfect square condition of the sums of opposite corners, so the latter is the solution.

Posted by Charlie on 2003-12-29 10:42:54