My son showed me three amazing dice the other day – Grime Dice. They are six-sided cubic dice but they don’t have the usual numbers 1 to 6 on each side. Each dice has a different combination of numbers which retain the same average value (3.5) as a normal dice.

The amazing property of these dice is discernible when you use them competitively – i.e. you roll one dice against another. If you roll each of them against a normal dice then as you might expect, each dice will win as often as it will lose. But if you roll them against each other something amazing happens.

Dice A will systematically beat Dice B

Dice B will systematically beat Dice C

and amazingly

Dice C will systematically beat Dice A

So the fact that Dice A beats Dice B, and Dice B beats Dice C does not ensure that Dice A will beat Dice C. Wow!

And how about this: If you ‘double up’ and roll 2 Dice A‘s against 2 Dice B‘s – the odds change around and now the B‘s will beat the A‘s ! Is that really possible? Well yes, and just to convince myself I wrote a Spreadsheet (.xlsx file) and generated the tables at the bottom of the article. If you download it you can change the numbers to try out other combinations.

There are lots of sets of non-transitive dice and they have many other surprising properties. This web page has more detail and the video below includes a chat with Mr. Grime himself.

I really don’t know what to make of these dice – but they did surprise me, so I thought I would share that surprise with you.

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Tags: Grime Dice, Maths, Non-transitive Dice