Nim is a great game to investigate in a range of contexts. Its accessible nature means that all children understand the rules and whilst they might make moves randomly to begin with, they will quickly build up a strategy. This version with money can be played either with subtraction or addition. I tend to prefer subtraction as it explicitly requires the exchanging of coins and notes. It works best if you use pretend coins.

We start with a £5 note. Players take it in turns to subtract either 50p or £1 from the money. The player that takes the last coin wins.

With games, it is important to give them a chance to just play the game and explore it first. On this Island, a lot of discoveries can be made from the starting point and as such, there are likely to be fewer individual explorations. That’s ok though! In developing a strategy, the children will use the Argue skills extensively. With a game like this, a good outcome could be to have a ‘beat the teacher’ contest. Can they come up with a viable strategy to beat you?

You could get them to use the Organise skill to create frequency tables about whether the first or second player wins.

To win, you want to leave the other player with multiples of £1.50. This is because you know you will win if you are left with either 50p or £1. Leaving the opponent on £1.50 ensures this and you can ensure you leave them on £1.50 if you leave them on £3 and so on. That means it is better to go first and take off 50p.

It’s easy to change the game a little by changing the starting amount.

**What if we started the game with £4?**

This doesn’t really change the strategy though. A good thought might be to get them to change the rules of the game so that the player going second can always win. It’s a nice collaborative exercise in mathematical thinking.

**What if we subtracted £1 or £2 each go?**

This extension deepens their understanding of the strategy. It is the same principle as before but with different coins. They can now generalise their strategy to work with different combinations of coins. In this, I’ve deliberately used two coins where one is double the other again as it connects to the initial exploration but other avenues are possible.

**What if we had three amounts to choose from?**

With an extra coin to choose from, each round will be longer but there will be more exchanging.

Once they understand the game and the strategy, exploring further always requires a twist to break what they thought was automatic.

**What if there were two piles of £5? **

Each turn you could choose which pile to take from. Can you manage to win both piles?

**What if you weren’t allowed to take the same amount twice in a row?**

This adds more complexity to the problem as the previously outlined strategy no longer works.

**Children’s possible use of reasoning skills (click on each heading):**

## Search

With it being a game and only having two options each time, there isn’t as much use of the Search skills for creating data. Early on in particular, there is going to be Roaming as they won’t know a strategy. Seeking might be used if they have a strategy and start changing the rules of the game and are doing actions that follow their strategy.

## Organise

It might be useful to collect whether or not the winner went first or second. At the start, the winner might be random but you can use this collection to ask them whether it is possible to make it one way or another and then collect this data again once they have a winning strategy.

## Discover

With the nature of the game, the conjectures are going to relate to their strategies. Ideally, what we want the children to do is link their conjectures to different versions of the game. Either trying to be more generalised or thinking about, before they try new rules, whether or not a previous conjecture will still apply to a new version.

## Investigate

A nice way to consider investigating in this inquiry is for them to think about how they could change the rules so that a conjecture/strategy no longer held true.

## Argue

Showing why a strategy works could be possible through some kind of tree diagram. Each node would be a decision to be made in a go. Through this, you could show that no matter what the other player did, you could always win if you left them with a certain amount.

## Explore

As mentioned in the Investigate skills section, the children can come up with their own ‘what if…?’ questions and you can add a problem solving element to it by making them consider how they can change the rules so that a certain player no longer always wins. The difficulty comes with them choosing coins that produce a result. If they start on £5 and the choice is between 1p and 5p, it isn’t going to be a very good game.

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