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Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

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Lesser Digits

How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

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Nines and Tens

Explain why it is that when you throw two dice you are more likely to get a score of 9 than of 10. What about the case of 3 dice? Is a score of 9 more likely then a score of 10 with 3 dice?

Coins

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?

This problem requires students to work systematically and apply some understanding of primes. It could also be adapted as a basis for work familiarising students with UK coinage.

Possible approaches

This task could be a suitable homework or "Problem of the Week" during a topic on Probability, Money or Primes. Solutions to the extension questions could be posted on a display board where students could check them and contribute further ideas over an extended period of time.

Alternatively the task could be used as an exercise in group work - give one member of the group a pen and paper, and give each of the others one (or two) clues. They must not show the clues to anyone or write them down. They can read clues out or explain them to the group. Each student is responsible that their clues are satisfied. The group needs to work together to find a solution to all the clues.

Key questions

Can you choose a good clue to start working on, one that gives you enough information to get started?
Are you considering ALL possibilities, or just trying to spot something that works?

Possible extension

This could be a good time to talk about the phrase "necessary and sufficient". Ask students to devise a set of clues that define a unique set of coins, and where no individual clue can be deleted without losing the uniqueness of the solution.

Possible support

Students could be given a pile of coins, (or a worksheet ofphotocopied/drawn coins) and asked to select sets of 5 coins from these, satisfying suitable clues - the total is...
an even number
one more than a multiple of 7
less than 10
all the coins are different, etc.

They could be asked to devise clues for each other, or to look for combinations of clues that define one specific set of coins.