Problem
Play the strategy game "Take Three or Less". Place twenty-one counters in a row. With a partner take turns, removing one, two or three counters each turn. The person to remove the last counter is the winner.
Is "Take Three" a fair game? [In a fair game, each player has an equal chance of winning.]

What is this problem about?
This problem is very similar to Take Two, Algebra Level 3. Hence the comments that we made there are equally as valuable here.

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.

Given that Take Two is an easier problem, you may want some of your class to tackle that one before this one.

Achievement Objectives
Algebra (Level 4)
- use a rule to make predictions.
Mathematical Processes
- devise and use problem solving strategies (draw a picture, use equipment, think).

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. Ask the children to clarify when a game is fair. (If there is an advantage in being the first or second player then the game is not fair.)
4. 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.
5. As the children play the game ask questions that focus their thinking on the patterns that they are using to find a winning strategy:
   What have you noticed in playing this game?
   If you are the first player how many counters should you take? Why?
   Do you prefer to be the first or second player? Why?
6. If the children are having problems looking for patterns suggest that they start with a smaller number of counters (7).
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, two or three. In this form of the game, the children start at 21 and take turns in subtracting one or two. Whoever can go to zero, wins.

Extension
Change the number of counters to 57 or any other number you prefer.
Change the number of counters that can be taken to one two, three or four.

Solution
A good problem solving strategy is to try a simpler problem. If your students haven’t done the problem Take Two (Level 3, Algebra) and they are having trouble with this present problem, then we suggest they try Take Two first.

The strategy here is very similar to that in Take Two. In Take Three, if only one, two or three counters left, then they can be removed for a win. On the other hand, four counters is a losing position. The next losing position is eight as a pile of eight can be reduced to four by the second player no matter what the first player does.

The losing numbers of counters are then twelve, sixteen and twenty. So the first player when faced with twenty-one, takes one counter away. Then the second player is on a losing number and should lose if the first player plays correctly. By ‘correctly’ here, we mean that, after the first move, if the second player takes x counters, the first player should take 4 – x, so that after these two moves, the pile is reduced by four to the next losing position.

This is therefore not a fair game as it is biased to the first player. On the other hand, any number of counters other than a multiple of four gives a win to the first player. If there are a multiple of four counters in play at the start, then the second player wins.