## Teaser 2825: Twin sets

**From The Sunday Times, 13th November 2016** [link]

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?

As stated there are multiple potential solutions to the puzzle. In the comments I give some additional conditions that allow a unique solution to be found (which is the same as the published solution).

[teaser2825]

## Jim Randell 9:42 am

on16 October 2019 Permalink |As originally stated there are multiple solutions to this puzzle.

I made the following additional suppositions in order to allow a unique solution to be arrived at.

The following Python program then finds the unique solution in 281ms.

Run:[ @repl.it ]Solution:The sum of Wunce’s squares is 1989. The sum of Repete’s squares is 831.Wunce’s list of squares is (324, 576, 1089) = (18², 24², 33²).

The sum of which is 1989. So in 2018 the twins would be 29.

Repete’s list of squares is (4, 9, 25, 36, 81, 100, 576) = (2², 3², 5², 6², 9², 10², 24²).

The sum of which is 831. Which is less than 841 = 29².

If we allow 1 in the list of squares then the sum of Repete’s list could be 832. And there are further possibilities if squares in the list are allowed to be repeated.

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