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 The digits and square numbers (Posted on 2006-10-12)
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 No Rating

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 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

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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