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If the base is

b, thenb²represented in basebis always 100. So this is the answer to 4 across.We are also told that 1 across and 3 down are palindromic numbers, so there are only 4 unknown values to place in the squares.

Additionally the smallest possible value for 1 down is 111. In base 18 this corresponds to 343 (decimal), which would make the son at least 114, so I don’t bother checking bases higher than this.

This Python program considers bases between 2 and 18 and creates a collection of alphametic expressions in that base, then uses the generalised alphametic solver (

SubstituedExpression()) from theenigma.pylibrary to look for solutions.It runs in 66ms

Solution:Your son is 37.The cross-figure is in base 7, and the filled out grid is:

1 across: 212 (base 7) = 107 (decimal), prime.

4 across: 100 (base 7) = 49 (decimal), 7².

5 across: 642 (base 7) = 324 (decimal), 18².

1 down: 216 (base 7) = 111 (decimal), 3× 37.

2 down: 104 (base 7) = 53 (decimal), prime.

3 down: 202 (base 7) = 100 (decimal), 10².

I used Jim’s analysis that the answer to 4 across is 100.

It is not very pretty code and I was not sure whether to post it, but it gets the answer in about 100 msec on my laptop. Fortunately, clue 3 down gives a unique answer for the number base, which is used for the rest of the code.