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The digits and square numbers (Posted on 2006-10-12) Difficulty: 3 of 5
All the nine digits are arranged here so as to form four square numbers.

9, 81, 324, 576

Which is the single smallest possible square number and a single largest possible square number using all the 9 digits exactly once?

What are the possible two, three & four number sets that follow this logic?

No Solution Yet Submitted by Salil    
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Solution computer solution (spoiler) | Comment 1 of 3

These are all the perfect squares using all 9 digits 1-9 exactly once, with no zeros:

139854276
152843769
157326849
215384976
245893761
254817369
326597184
361874529
375468129
382945761
385297641
412739856
523814769
529874361
537219684
549386721
587432169
589324176
597362481
615387249
627953481
653927184
672935481
697435281
714653289
735982641
743816529
842973156
847159236
923187456

There are 30 of them.
 
The smallest is 139854276
The largest is 923187456

All pairs of squares of the sort are:

 324  751689
 3249  15876
 36  5184729
 36  5948721
 4356  71289
 576  321489
 576  349281
 576  381924
 729  385641
 81  2537649
 81  5673924
 81  7436529
 81  9253764
 8649  35721
 9  13527684
 9  34857216
 9  65318724
 9  73256481
 9  81432576
 
 The triplets are:
 
 1  256  73984
 1  4  3297856
 1  4  3857296
 1  4  5827396
 1  4  6385729
 1  4  8567329
 1  4  9572836
 1  49  872356
 1  625  73984
 16  25  73984
 1  64  537289
 16  784  5329
 25  784  1369
 25  784  1936
 25  841  7396
 361  529  784
 36  729  5184
 36  81  74529
 36  81  79524
 4  16  537289
 4  25  139876
 4  25  391876
 4  289  15376
 81  324  7569
 81  576  3249
 81  729  4356
 9  324  15876
 
 The quadruplets:
 
 1  36  529  784
 1  4  9  872356
 4  25  81  7396
 9  25  361  784
 9  81  324  576
 
 ... and theres a set of five:
 
 1  9  25  36  784
 
 There is no set of six.
 
 DECLARE SUB test3 (s$, s1#, s2#, s3#)
 DECLARE SUB test4 (s$, s1#, s2#, s3#, s4#)
 DECLARE SUB test5 (s$, s1#, s2#, s3#, s4#, s5#)
 DECLARE SUB test6 (s$, s1#, s2#, s3#, s4#, s5#, s6#)
 DECLARE FUNCTION isSq# (x#)
 DECLARE SUB permute (a$)
 DEFDBL A-Z
 OPEN "digandsq.txt" FOR OUTPUT AS #2
 large$ = " ": small$ = "z"
 FOR i = 11111 TO 31427
  n$ = LTRIM$(STR$(i * i))
  good = 1
  IF INSTR(n$, "0") THEN
   good = 0
  ELSE
    FOR j = 1 TO 8
     IF INSTR(MID$(n$, j + 1), MID$(n$, j, 1)) THEN good = 0
    NEXT
  END IF
  IF good THEN
    PRINT #2, n$: sCt = sCt + 1
    IF n$ > large$ THEN large$ = n$
    IF n$ < small$ THEN small$ = n$
  END IF
 NEXT
 PRINT #2, sCt: PRINT #2, small$: PRINT #2, large$
 
 PRINT #2,
 
 
 n$ = "123456789"
 h$ = n$
 
 DO
  n1 = VAL(LEFT$(n$, 1)): n2 = VAL(MID$(n$, 2))
  IF isSq(n1) AND isSq(n2) THEN PRINT #2, n1; n2
  n1 = VAL(LEFT$(n$, 2)): n2 = VAL(MID$(n$, 3))
  IF isSq(n1) AND isSq(n2) THEN PRINT #2, n1; n2
  n1 = VAL(LEFT$(n$, 3)): n2 = VAL(MID$(n$, 4))
  IF isSq(n1) AND isSq(n2) THEN PRINT #2, n1; n2
  n1 = VAL(LEFT$(n$, 4)): n2 = VAL(MID$(n$, 5))
  IF isSq(n1) AND isSq(n2) THEN PRINT #2, n1; n2
  permute n$
 LOOP UNTIL n$ = h$
 
 DO
  test3 n$, 1, 1, 7
  test3 n$, 1, 2, 6
  test3 n$, 1, 3, 5
  test3 n$, 1, 4, 4
  test3 n$, 2, 2, 5
  test3 n$, 2, 3, 4
  test3 n$, 3, 3, 3
  permute n$
 LOOP UNTIL n$ = h$
 
 testing4:
 DO
  test4 n$, 1, 1, 1, 6
  test4 n$, 1, 1, 2, 5
  test4 n$, 1, 1, 3, 4
  test4 n$, 1, 2, 2, 4
  test4 n$, 1, 2, 3, 3
  test4 n$, 2, 2, 2, 3
  permute n$
 LOOP UNTIL n$ = h$
 
 
 testing5:
 DO
  test5 n$, 1, 1, 1, 2, 4
  test5 n$, 1, 1, 1, 3, 3
  test5 n$, 1, 1, 2, 2, 3
  test5 n$, 1, 2, 2, 2, 2
  permute n$
 LOOP UNTIL n$ = h$
 
 testing6:
 DO
  test6 n$, 1, 1, 1, 2, 2, 2
  permute n$
 LOOP UNTIL n$ = h$
 
 CLOSE
 
 FUNCTION isSq (x)
  t = INT(SQR(x) + .5)
  IF t * t = x THEN isSq = 1:  ELSE isSq = 0
 END FUNCTION
 

 DEFDBL A-Z
 SUB test3 (s$, s1, s2, s3)
   n1 = VAL(LEFT$(s$, s1))
   n2 = VAL(MID$(s$, s1 + 1, s2))
   n3 = VAL(MID$(s$, s1 + s2 + 1, s3))
   IF n1 < n2 AND n2 < n3 THEN
     IF isSq(n1) AND isSq(n2) AND isSq(n3) THEN
        PRINT #2, n1; n2; n3
     END IF
   END IF
 END SUB
 
 SUB test4 (s$, s1, s2, s3, s4)
   n1 = VAL(LEFT$(s$, s1))
   n2 = VAL(MID$(s$, s1 + 1, s2))
   n3 = VAL(MID$(s$, s1 + s2 + 1, s3))
   n4 = VAL(MID$(s$, s1 + s2 + s3 + 1, s4))
   IF n1 < n2 AND n2 < n3 AND n3 < n4 THEN
     IF isSq(n1) AND isSq(n2) AND isSq(n3) AND isSq(n4) THEN
        PRINT #2, n1; n2; n3; n4
     END IF
   END IF
 
 END SUB
 
 SUB test5 (s$, s1, s2, s3, s4, s5)
   n1 = VAL(LEFT$(s$, s1))
   n2 = VAL(MID$(s$, s1 + 1, s2))
   n3 = VAL(MID$(s$, s1 + s2 + 1, s3))
   n4 = VAL(MID$(s$, s1 + s2 + s3 + 1, s4))
   n5 = VAL(MID$(s$, s1 + s2 + s3 + s4 + 1, s5))
   IF n1 < n2 AND n2 < n3 AND n3 < n4 AND n4 < n5 THEN
     IF isSq(n1) AND isSq(n2) AND isSq(n3) AND isSq(n4) AND isSq(n5) THEN
        PRINT #2, n1; n2; n3; n4; n5
     END IF
   END IF
 
 END SUB
 
 SUB test6 (s$, s1, s2, s3, s4, s5, s6)
   n1 = VAL(LEFT$(s$, s1))
   n2 = VAL(MID$(s$, s1 + 1, s2))
   n3 = VAL(MID$(s$, s1 + s2 + 1, s3))
   n4 = VAL(MID$(s$, s1 + s2 + s3 + 1, s4))
   n5 = VAL(MID$(s$, s1 + s2 + s3 + s4 + 1, s5))
   n6 = VAL(MID$(s$, s1 + s2 + s3 + s4 + s5 + 1, s5))
   IF n1 < n2 AND n2 < n3 AND n3 < n4 AND n4 < n5 AND n5 < n6 THEN
     IF isSq(n1) AND isSq(n2) AND isSq(n3) AND isSq(n4) AND isSq(n5) AND isSq(n6) THEN
        PRINT #2, n1; n2; n3; n4; n5; n6
     END IF
   END IF
 
 END SUB
 

The permute subroutine is shown elsewhere on the site.


  Posted by Charlie on 2006-10-12 11:35:28
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