# Sunday Times Teaser 3130 – Making Squares

*by Andrew Skidmore*

#### Published Sunday September 18 2022 (link)

Liam has nine identical dice. Each die has the usual numbers of spots from 1 to 6 on the faces, with the numbers of spots on opposite faces adding to 7. He sits at a table and places the dice in a 3×3 square block arrangement.

As I walk round the table I see that (converting numbers of spots to digits) each vertical face forms a different three-figure square number without a repeating digit.

As Liam looks down he sees six three-digit numbers (reading left to right and top to bottom) formed by the top face of the block, three of which are squares. The total of the six numbers is less than 2000.

What is that total?

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My code finds several solutions including the one yours finds. I’m not sure where I went wrong. Please excuse the unformatted printout.

Hi John,

I don’t yet know if it is the cause of your multiple solutions, but I noticed that you are not using identical dice.

Dice come in two forms – right-handed (western) ones have the 1, 2, 3 going anti-clockwise around the vertex while on left-handed (eastern) dice the 1, 2, 3 go clockwise around the vertex.

The other issue is that fixing one side and permuting three in lines 16/17 doesn’t work because the rotational invariance is broken by the way the top numbers are read out. Don’t be too hard on yourself here as both Frits and I made the same mistake first time round.

‘identical dice’ of course! Here is my revised solution