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Can you explain the strategy for winning this game with any target? Counting Factors

Is there an efficient way to work out how many factors a large number has?

Ben's Game

Why do this problem?

This problem requires some simple knowledge of fractions and multiples and demands some strategic thinking. It may offer a good opportunity to compare methods between students - there isn't just one route to the solution. Note that there is no need to use algebra in this problem.

Possible approach

This printable worksheet may be useful: Ben's Game.

Choose three students to act out the scenario with a (real or imaginary) pot of $40$ counters, as described under support below.

Ask students to work in pairs or small groups to try and find the answers. If any groups are successful too quickly (!) ask them to change one or more of the fractions, and to adapt their strategies to the new situations.

As a group discuss the methods used.
What worked? What didn't work?
If faced with a similar problem in future, which methods would the class use?

Methods students have used include:
• trial and error, making strong use of the upper bound of 40,
• working backwards from the fact that they end up with the same amount.
• focusing on one individual child's initial share of counters
• use the fact that each child passes a certain fraction to their neighbour.
• algebraic representation
Because the problem has many variables, students will need to devise a clear recording system

Key questions

Can Ben start with $10$ counters?
Why can't they use all $40$ counters in this game?
What are the possibilities for Emma's first pile of counters?
What possible numbers of counters could each child end up with?
Will there be just one solution? No solution? Many solutions?

Possible extension

Change the numbers. What if the $40$ were $100$, or a limitless supply of counters?
What if the third, quarter and fifth were different fractions - unitary or not?
What generalisations can you find in the solutions?

Possible support

Group students in $3$s and provide sets of $40$ counters. Ask them to play this game as a group and challenge them to be the first group to find the answer. Use these rules
1. Decide who will be Emma, Jack and Ben.
2. Emma chooses any number of counters from the $40$ counters and notes this number down
3. Jack then chooses any number of counters from those left over and notes this number down
4. Ben then chooses any number of counters from those left over and notes this number down
5. Emma, Jack and Ben then find a fifth, quarter and third of their counters respectively
6. All pass the counters to their neighbour
7. If they all have the same number then the group has won.
As students try to play this game they will encounter difficulties in making the fractions and see that the number of counters taken must be quite specific in order to end up with the same amount at the end. Students should be encouraged to learn from their mistakes to try to find out the winning combination.