# Board games and mathematics: trivial pursuits

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- Board games and mathematics: trivial pursuits

## Board games and mathematics: trivial pursuits

Can you shuffle a deck of cards with a single card in it?

In the board gaming world, this question is particularly interesting, because considering an answer for it draws out the two meanings of the world ‘shuffle’ – one, meaning to move cards around physically in order to make them more random, and two, meaning to go through a largely abstract process which may be part of a game rule.

To try and explain the second meaning in more detail: If we said that you ‘couldn’t’ shuffle a deck of cards with only a single card in it, and the rules of the game stated ‘if you shuffle your deck, you may then complete X action’, we might consequently conclude we weren’t allowed to do X action, which would usually be inconsistent with most board game play.

In mathematics, we might call the single-card shuffle ‘trivial’. This is the word we tend to reserve for ‘stuff that is really obvious but often a good starting point for solving problems’, which is interesting, because of course what constitutes ‘obvious’ depends on your level of mathematical knowledge, intuition and experience. Some of the best ‘trivial’ ideas in mathematics are actually philosophical in nature, and we’d probably include this card question in there too, as a nice example of a crossover question between mathematics and philosophy – a space where some of the simplest of questions have the most complicated of thoughts associated with them. (This question feels a little like the philosophical question: ‘Can one hand clap?’)

So ‘trivial’ is likely quite the misnomer, and far too dismissive for such weighty and important kinds of thinking. If we build solutions, thought experiments and problem-solving strategies on top of such foundations, surely they should be called ‘substantial’ instead?

There is another nice parallel here with mathematics. We gave two possible meanings to the idea of ‘shuffle’, which related to the physical act of shuffling and the more abstract or algorithmic sense of the verb. In mathematics education, we often consider mathematical processes as ‘concrete’ or ‘abstract’ (or more likely, some hybrid) in a similar vein. What does this mean? For example, to ‘scale up’ something might mean to draw it physically bigger on the page – but we can also achieve this by a more abstract process of multiplication, which can be done completely independently of the physical manipulatives themselves, and which also has a separate abstract meaning. Can we ‘scale up’ by a scale factor of 1, and what does this mean? Would you call this result ‘trivial’ or ‘substantial’, and why?

Both this example and the single card shuffle have an additional problem: that of definition. In philosophy, it is almost always possible to create a definition for something which breaks its own system. If you try to define a rule for what shuffling ‘is’, you might end up with something paradoxical. For example, if we describe the arrangement of the cards as a single state in a set of possible states, we could define shuffling as the act of changing the cards to a new state and declare that the one card shuffle is impossible. But it is perfectly possible to permute a larger deck of cards randomly and end with them in the same state that they started, suggesting that despite a process of rearrangement having taken place, the deck has not in fact been ‘shuffled’ by this definition. Clearly this doesn’t make intuitive sense, and it feels like the same sort of idea as scaling up by a scale factor of 1, or performing an identity operation (a ‘process’ that leaves the thing unchanged). They certainly seem connected. The question is, have we in fact stumbled upon a ‘trivial’ metaphor for the limitations of axiomatic systems captured in Godel’s incompleteness theorems?

Or a substantial one?

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