Problem
Play the strategy game "take two".
Place five counters in a row. With a partner take turns, removing one or two counters each turn. The person to remove the last counter is the winner.

1. Can you find a game strategy so that the first player always win?
2. Is this a fair game? [In a fair game, each player has an equal chance of winning.]

What is this problem about?
First it is worth noting that this problem is an open one. We don’t tell you whether the game is fair or not. If it is fair, then it doesn’t matter who starts, each player is equally likely to win. On the other hand if the game is not fair, then you have to decide who will win, the first player or the second player.

Open problems like this are among the more difficult because you have to decide which direction to go. Generally this is not easy and you only figure out which answer to look for after playing the game many times. Beware jumping to a quick conclusion on the basis of one or two games.

In the real world and in the world of research mathematics, open problems occur all the time. Hence it is important to have some experience with them. Remember, to solve them requires a lot of experimentation. Consequently it might be a good idea first of all for the class to play this game several times and then pool the results. At this stage you might have enough evidence to go in the right direction.

The second thing worth noting here is that this is a problem about patterns in disguise. By following a sequence, the right strategy can be found.

Achievement Objectives
Algebra (Level 3)
- describe in words, rules for continuing number and spatial sequence patterns;
- make up and use a rule to create a sequential pattern.
Mathematical Processes
- devise and use problem solving strategies (draw a picture, use equipment, think, work backwards).

Resources
- Counters
- Copymaster of the problem (English)

Specific Learning Outcomes
The children will be able to:
- Select and use a problem solving strategy (eg use equipment, work backwards, draw a picture).
- Identify patterns used in solving the problem (multiples of 3).

1. Introduce the problem by playing one game with the class.
2. Read the problem.
3. Let the children play the games in pairs. It is important to stress the idea that they are playing the game together to see if they can work out a winning strategy for the first player. By doing this you are encouraging them to analyse the game rather than just trying to beat their opponent.
4. As the children play the game ask questions that focus their thinking on the patterns that they are using to solve the game:
5. 
   What have you noticed in playing this game?
   If you are the first player how many counters should you take? Why?
6. If the children are having problems looking for patterns suggest that they start with 3 counters.
7. When the children think that they have a strategy for "winning" the game let them try their strategy out with another pair. (At this stage ask the children to keep their ideas to themselves.)
8. Once the pairs have played a couple of games ask them to share and discuss their ideas with the other pair. Encourage the group of 4 to write down their method for "winning" the game and their ideas about whether the game is fair or not.
9. Share strategies for playing the game.
10. Discuss: Do you think that the game is fair? Why or Why not?

Other Contexts
Of course the counters can be money or any object. But you can get the children to do this in their heads using numbers. As such it is a good exercise in mental arithmetic for subtracting one or two. In this form of the game, the children start at 7 and take turns in subtracting one or two. Whoever is forced to take away the last one, loses.

Extension
Change the number of counters to 7 or any other number you prefer.

Solution
This is a nice opportunity to work backwards. The person who wins is the one who takes the last counter or the last two counters. Let’s call this person, person A. So the person, person B, who goes just before this will have had three counters in front of them. (If they had had two, they would have taken away the two and have won. If they had had four, they would have taken away one and left three and so put themselves in the winning position or have taken away two and left two – a losing position.)

So three is a losing position. The next highest losing position is six. This is because if person B sees six counters then B can only take one or two counters away to reduce the pile to five or four. Then person A can take two or one counters to reduce the pile to three and put B in a losing position.

Now there were five counters originally. The first person who plays is the only one who can get the pile down to three and put the second person in a losing position. So the first player can always win. So the game is not fair.

It’s worth noting that the first player will also win if there are four counters. The first player takes one counter and reduces the pile to three. This is a losing position for the second player.

But sic counters in the original pile means a winning game for the second player. No matter what the first player does, the second player reduces the pile to three and wins from there.

In general, if the number of counters originally was a multiple of three, the first person will lose if the second player knows how to play the game. On the other hand if the number of counters is not a multiple of three, then the first player wins by reducing the pile to a multiple of three and making sure that each time he plays the pile is reduced to a smaller multiple of three.

We show the first person’s strategy for an 11 counter game in the table below.