Solution to Enigma No. 1518

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Answer: (1) 855 (2) 1989

The winner is Michael Horner of Horowhenua, New Zealand. There were 176 entries.

Worked answer

Squares with no repeated digits that use:

3: 36, 324, 361, 1369…

5: 25, 256, 529, 576, 625, 3025…

7: 576, 729, 784, 1764…

0: 1024, 1089, 2304…

(1)

The smallest square to use 7 is 576, but that prevents the use of 36, and for 3 we have to use 324, with 1, 8 and 9 still to be used: 576 + 324 + 81 + 9 = 990.

729 prevents the use of any 2-digit or 3-digit square for 5; but with 784 we may use both 36 and 25, with 9 and 1: 784 + 36 + 25 + 9 + 1 = 855 – so there is no need to check any further squares that use 7.

(2)

The smallest square to use 0 is 1024: that leaves 576 as the only 3-digit square available that uses 7. 3, 8 and 9 are still to be used, and there is no solution.

Rather than try 1024 with a 4-digit square that uses 7, it is better to try 1089. That too leaves 576 as the only 3-digit square available that uses 7. 2, 3 and 4 are still to be used, and 324 is a perfect square: 1089 + 576 + 324 = 1989 – so there is no need to check any further squares that use 0 or 7.

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