## Teaser 3130: Making squares

**From The Sunday Times, 18th September 2022** [link] [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?

[teaser3130]

## Jim Randell 5:29 pm

on16 September 2022 Permalink |At first I found multiple solutions. But you can find a unique answer to the puzzle if you ensure the dice really are

identical.This Python program runs in 261ms.

Run:[ @replit ](or a faster variation on [@replit]).

Solution:The total is 1804.The dice are laid out as follows:

So the total is: (115 +

441+ 445) + (144+144+ 515) = 1804.Of the six numbers read off from the top of the arrangement the square numbers are: 441 (= 21²) and 144 (twice; = 12²).

Note that each of the corner dice is

left-handed(i.e. a mirror image of a “standard” die), and so, as the dice are allidentical, they must all be left-handed.If we are allowed to mix left- and right-handed dice, then there are many possible layouts (and many possible answers to the puzzle).

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## Frits 9:50 pm

on16 September 2022 Permalink |Thanks to Jim, hopefully with the correct solution.

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## Frits 10:03 pm

on16 September 2022 Permalink |The side squares should be seen when walking anti-clockwise around the table (so the top and right squares are printed reversed).

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## Hugh Casement 7:17 am

on17 September 2022 Permalink |Are they left-handed dice? I can’t find any solution with standard dice (such as Frits shows).

Or perhaps you have to walk round the table with your head upside down.

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## Jim Randell 7:25 am

on17 September 2022 Permalink |@Hugh: Yes. There is only a solution if left-handed dice are used (at least in the corners of the layout – and as the dice are identical then the rest must be left-handed too).

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## Frits 9:40 am

on17 September 2022 Permalink |I set up my dice right-handed (I didn’t even know the term right-handed) based on numbers facing up when going clock wise. However, the corner arrangements in the block have to be read anti-clockwise so I should have used the 7-complement of my hardcoded values.

My solution is the same as Jim’s and is left-handed. I will post a new version checking both left-handed and right-handed dice (using Brian Gladman’s function for determining the hardcoded values).

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## Frits 1:52 pm

on17 September 2022 Permalink |Supporting right-hand and left-hand dice and with a recursive version of Brian Gladman’s function for third face values.

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## Frits 11:39 pm

on17 September 2022 Permalink |Based on Jim’s first posted program. This program runs in 90ms.

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## Frits 11:02 pm

on22 September 2022 Permalink |More efficient version.

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