This is a game for two players. You will need some small-square
grid paper, a die and two felt-tip pens or highlighters. Players
take turns to roll the die, then move that number of squares in a
straight line. Move only vertically (up/down) or horizontally
(across), never diagonally. You can cross over the other player's
trails. You can trace over the top of the other player's trails.
You can cross over a single trail of your own, but can never cross
a pair of your trails (side-by-side) or trace over your own trail.
To win, you must roll the exact number needed to finish in the
target square. You can never pass through the target square. The
game ends when a player ends his/her trail in the target square, OR
when a player cannot move without breaking any of the rules.
Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?
Age 11 to 14 Challenge Level:
Why do this problem?
This problem offers an opportunity to explore and discuss two types of probability: experimental and theoretical. The simulation generates lots of experimental data quickly, freeing time to focus on predictions, analysis and justifications.
Invite children to make hypotheses about the likelihood of winning with two discs. Encourage pupils to explain their thinking and try to justify their hypotheses.
Run the simulation, once, then again and again, so that pupils are confident with what the computer is doing. Then run it 100 times and ask the pupils to comment on/explain the result. (Clicking on the horizontal panel at the bottom of the right-hand window changes the recording method from a graph to a table.)
Record the result for two discs on the board before moving onto three discs.
Invite pupils to think for a minute on their own about what will happen with three discs. Ask them to justify their conjectures with a partner and then the whole class. Use the interactivity to demonstrate what happens. Pupils can then work, perhaps in pairs, to explain the results.
You may like to stop the group part way through their work in order to talk about some effective recording methods that pairs have devised.
When ready pupils can move on to four, five, six and n discs.
How can you decide if a game is fair?
How many goes do you think we need to be confident of the likelihood of winning?
Are there efficient systems for recording the different possible combinations?
Can you justify any general findings you have made?
How can we colour three or more discs so that we have an even chance of winning?
If we retain the current colouring of the discs, how can we change the cirteria for winning so that we have an even chance of being successful?
Students can use coins or coloured counters to help them list all the possible outcomes.
Teachers may want to use this recording tool to gather the results of other similar experiments that their students are carrying out:
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice.