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More Number Sandwiches

Age 11 to 16 Challenge Level:

Why do this problem?

This problem follows on from Number Sandwiches for those students whose curiosity has been sparked by the initial problem and who are desperate to know "What if...?"
Having explored 3-, 4- and 7-sandwiches, the natural question that arises is "Can I make 5- and 6-sandwiches?" 

Possible approach

Once students have had plenty of time to work on the original problem, some students will perhaps wonder why you have not asked them to explore 5- and 6-sandwiches. The interactivity is coloured red and blue to draw students' attention to what happens when you place a pair of odd numbers, or even numbers, on the grid.

We hope that this, together with the key questions below, will help students to construct a proof for the impossibility of 5- and 6- sandwiches, and perhaps generalise to the other impossible sandwiches.

You can see an outline of the proof in the article Impossible Sandwiches

Key questions

In a "7-sandwich", how many red squares are covered and how many blue squares are covered?
If it were possible to make a "6-sandwich", how many red squares and how many blue squares would be covered?
If you place a 1 on a blue square, on which colour will you place the other 1?
If you place a 2 on a blue square, on which colour will you place the other 2?
If you place a 3 on a blue square, on which colour will you place the other 3?...
In general, what can you say about the colours on which you place pairs of numbers?

When you try to make a sandwich with the numbers from 1 to 5, or from 1 to 6, what goes wrong?

Possible extension

Students who know about computer programming may like to write a program to find all the 7-sandwiches, or explore higher order sandwiches.

Possible support

Encourage students to work systematically to rule out possibilities for the 5-sandwich, in order to provoke the realisation of why it is impossible.