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Sandwiches

Stage: 2, 3, 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem?
The Sandwiches problem offers a challenge for everyone at all levels. It is particularly valuable in primary schools because it gives young learners an opportunity to work out a proof and to explain it. They can find the solution for the 3-sandwich (consisting of 1's, 2's and 3's) and discover that it is impossible for the 2-sandwich. Even very young learners can explain why (prove that) it is impossible to find a solution for the 2-sandwich. You may decide to stop at 3-sandwiches or try to find 4-sandwiches as well.

At upper primary and lower secondary level, as there are many solutions in the case of 7-sandwiches and 8-sandwiches, the problem provides an opportunity for many individual learners to have success in discovering their very own solution, different to any that have already been found.

Having investigated 2, 3, 4 and 7-sandwiches, if learners in your class have been encouraged to ask "What if..." and look for generalisations, the natural question is "What about 5-sandwiches and 6 sandwiches?"

The experience of learning to think mathematically offered by this problem is equally valuable to learners in upper secondary school.

Possible approach
It is helpful, particularly for young learners, to have digits to rearrange (either plastic or simply written on paper or card).

It is a good idea to have a 'Challenge-Chart' on the classroom wall where new solutions can be written up as people discover them.

As there are many solutions for n=7, this problem calls for you to work systematically in order to find them all.

Key question
What sandwiches can you make?
Can you make 2-sandwiches and if not why not?
Are any sandwiches the same looked at in different ways?
Is it possible to make 5-sandwiches?
For which values of n can n-sandwiches be made and for which values of n is it impossible? Why?

Possible support
You may want to stop with n=4.

Possible extension
The process of working systematically through all possible cases can be tedious and time consuming. This is where discussion by the class of the best systematic approach, and then sharing the work of searching for solutions and checking, offers a valuable learning experience. Such work prepares learners for using computers in problem solving and not just in mathematics but also in other fields.

In order to construct a computer program you have first to plan a systematic approach to the problem. Another extension could be to write a computer program to find solutions and to test which values of n yield n-sandwiches.

Finally there is a method of proving that for certain values of n it is impossible to make n-sandwiches. The argument is simple and requires only very elementary mathematics. The proof falls into the 'Aha' category - once you see it, it seems obvious and amazingly simple, but the choice of method called for real inspiration in the first place. This is an existence proof not a construction proof - it does not give a method of constructing solutions. See the article Impossible Sandwiches.