Sunday Times Teaser 2825 – Twin Sets

by Victor Bryant

The twins Wunce and Repete each made a list of positive perfect squares. In Wunce’s list each of the digits 0 to 9 was used exactly once, whereas in Repete’s list each of the digits was used at least once. Wunce commented that the sum of his squares equalled their year of birth, and Repete responded by saying that the sum of his squares was less than the square of their age .

What is the sum of Wunce’s squares, and what is the sum of Repete’s?

 “.. in 2018.” was omitted in error here.

1. There are problems with the wording of this teaser beyond that noted above. Repete’s list of squares could be [4, 9, 25, 36, 81, 100, 576] with a sum of 831. But we can add 1’s, 4’s or a 9 without exceeding the 841 sum of squares limit set by age of 29. Even if we assume that the lists don’t contain duplicates, we can still add a one to Repete’s set of squares so we don’t know whether his sum is 831 or 832.

2. I’ve used the following clarifications to give a unique solution to this puzzle:

1. the puzzle is set on/after the twins birthday in 2018;
2. the lists are of *different* squares;
3. the squares are greater than 1

With these clarifications this Python 3 program finds the solution in 281ms.

The year (Y) and minimum value to be squared (m) can be specified on the command line.

If m=1 is specified there is an additional solution for Repete.

Another way to fix the puzzle is to specify that the digits in Repete’s list appear either once or twice, but that requires code changes to implement.

As usual, the latest version of the enigma.py library is available at [ https://www.magwag.plus.com/jim/enigma.html ].