This is an open problem presented as a game that involves decisions and fairness. Students learn that if it is fair, then it doesn’t matter who starts and each player is equally likely to win. If the game is not fair, then a decision must be made about who will win, the first player or the second player. By playing a number of games students have the opportunity to begin to see a pattern, and to identify a strategy.
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.
- Can you find a game strategy so that the first player always win?
- Is this a fair game? [In a fair game, each player has an equal chance of winning.]
- Introduce the problem by playing one game with the class.
- Read the problem.
- Let the students play the games in pairs. It is important to stress 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.
- As the students play the game ask questions that focus their thinking on the patterns that they are using to solve the game:
- What have you noticed in playing this game?
If you are the first player how many counters should you take? Why?
- If the students are having problems looking for patterns suggest that they start with 3 counters.
- When the students think that they have a strategy for "winning" the game let them try their strategy out with another pair. (At this stage ask the students to keep their ideas to themselves.)
- 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.
- Share strategies for playing the game.
- Discuss: Do you think that the game is fair? Why or Why not?
Extension to the problem
Change the number of counters to 7 or any other number you prefer.
This is an opportunity to work backwards. The person who takes the last counter or the last two counters is the winner. (Person A.) 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.
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 six 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 they play 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.
|counters in pile||11||9||8||6||4||3||1|
|multiple of three||9||8||6||3||1||first player wins|