*Spoiler for New Scientist Enigma 1703:*

I was staying at my sister’s house when my niece Amy came home from school feeling special. The class had been shown how to split a whole number, T, into whole number parts in such a way that the product of the parts was the greatest, G, that could be obtained for that T. For instance, she explained, 10 could be split into ten ones, or 2 and 4 and 4, or 5 and 5, and so on, which would yield products of 1, 32, and 25 respectively. But, she warned, G exceeds 32 for T=10.

Why did she feel special? Well, each pupil in the class had been given a different number in the range 20-50 inclusive for their personal T, and she had noted that, when she added the digits of her G together, the sum was exactly half of her T, and no one else in the class had T and G with this property.

What value of T was Amy given?

I couldn’t do this without a good deal of computer help. I wrote a Perl script to find *G*(*T*) for small values of *T*; the number of possible partitions of *T* becomes too large to test all the numbers out to 50, but it did show me the pattern, which once you know it can be proved by induction:

- For
*T*= 3*k*,*G*(*T*) = 3.^{k} - For
*T*= 3*k*+1,*G*(*T*) = 4 x 3^{k–1}. - For
*T*= 3*k*+2,*G*(*T*) = 2 x 3.^{k}

For instance, for *T* = 3, *k* = 1 and G(3) = 3; for *T* = 4, *k* = 1 and G(4) = 1 x 3 = 3; for *T* = 5, *k* = 1 and G(3) = 2 x 3 = 6; for *T* = 6, *k* = 2 and G(3) = 3 x 3 = 9, and so on.

That’s about as interesting as this enigma gets. After that you pretty much just have to calculate *G*(*T*) for all even *T* from 20 to 50 and find the one whose digits sum to half of *T*. More computer work; I wrote a spreadsheet. The answer is *T* = 36, *G*(*T*) = 3^{12} = 531441 with digit sum 18.