Spoiler for New Scientist Enigma 1695:

In the following list of five numbers I have consistently replaced digits by letters, with different letters being used for different digits: CAST THE ODD ONE OUT. All but one of those five numbers is a perfect square. What is the numerical value of the odd one out?

At least two of THE, ONE, and OUT are squares with no repeated digits, so we need a list of those:

169, 196, 256, 289, 324, 361, 529, 576, 625, 729, 784, 841, 961

ODD may be a 3-digit square with last two digits the same: 100, 144, 400, 900

If ODD is 100 or 144 then one of ONE and OUT must be 169 or 196 while the other is not a square. If ODD is 400 then neither ONE nor OUT can be a square, so ODD isn’t 400. If ODD is 900 then one of ONE and OUT must be 961 while the other is not a square.

Therefore either ODD is not a square, or it is and one of ONE and OUT is not. Either way, CAST and THE are squares.

T is the last digit of a square (that does not end with the same digit twice) so must be 1, 4, 5, 6, or 9. Therefore THE can be only 169, 196, 529, 576, 625, or 961.

Suppose ODD is not a square. Then possible values for ONE and OUT are: 256 and 289 (in either order), 324 and 361, 529 and 576, or 729 and 784. Then the first and last digits of THE, in either order, must be 6 and 9, 1 and 4, 6 and 9, or 4 and 9, respectively. But none of the possible values of THE satisfies any of those requirements. Therefore ODD is a square, and either ONE or OUT is the non square.

If ODD is 900 then one of ONE and OUT must be 961, so T is 1 or 9; but then H is 6 or 9, both of which have been used.

If ODD is 100 or 144 and OUT is 169 then THE must be 961, re-using a digit.

If ODD is 100 or 144 and OUT is 196 then THE must be 625. CAST must end in 6 and the four unused digits are 0, 3, 7, 8 or 3, 4, 7, 8 depending on ODD. Either way there are no 4-digit squares ending in 6 with the other 3 digits different and chosen from 0, 3, 4, 7, 8.

Similarly, if ODD is 100 or 144 and ONE is 169 then THE must be 529, and there are no values for CAST.

If ODD is 100 and ONE is 196 then THE must be 576, and there are no values for CAST.

The one remaining possibility is: ODD is 144, ONE is 196. THE must be 576 and CAST must be 3025. By elimination U is 8 and CAST THE ODD ONE OUT is 3025 576 144 196 185.

The non square is 185.

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*Related*

You can add that T cannot equal 4, since all the squares beginning with 4 i.e. 400,441,484 all have repeated digits and THE is three different digits.