## Enigma 1437 – Two Squared

*set by Richard England*

### Question

Harry and Tom each replaced each asterisk with a digit in such a way that the three numbers that could be read across and the two numbers that could be read down were five different perfect squares each of which consisted of three different digits (with no leading zero). Tom did not use any of the 3-digit squares that Harry used.

Which three 3-digit squares (of three different digits with no leading zero) did neither of them use?

### Answer

First lets get all the 3 digits perfect squares (of which there are 13). They are as follows:

169, 196, 256, 289, 324, 361, 529, 576, 625, 729, 784, 841, and 961 *(corresponding to squares of integers 13 – 31)*

We could just put these numbers in the format and test them, so that’s what I did first.

After some trail and error I came upon a solution.

These two worked fine but wait. I turns out there are three possible ways of arranging the numbers.

and

After comparing all three of these we can see that they only use the same 10 squares out of the thirteen. So the three digits that the were not used turned out to be 324, 729, 784. And just for the hell of it, here’s a graphic for that too.

Oh the fun. We can only hope they’re going to get harder some time soon.

*(On a side note, as I don’t always get the time early in the week to solve the puzzles as they are set so instead of holding back posting the complete question/solution when I have, I’m going to post up the question when I get them – usually Thursday/Friday – and update them when I’m happy I have a solution. If anybody fancies a crack at them please feel free.)*