### Why do this problem?

At the heart of

this
problem is using a representation which helps to simplify the
situation. For example, in the first part, using the fact that any
number is either odd or even reveals the structure. The context and
argument can be based on modulo arithmetic. Working from specific
cases in order to generalise is a mathematical technique that can
also be highlighted.

### Possible approach

You might like to look at the problem

Make
37 first.

Ask the group to give you three numbers and write them on the
board, in a second column write their three differences and in the
third column the product.

Repeat several times until you are ready to discuss any
patterns and relationships the group have noticed. Particularly
that the product is always even.

- Can they find three numbers where this is not the case?
- Why not?

Discuss the partitioning of integers into odds and even
numbers. For example:

All numbers are either even (E)

XX...XXXXXXXX

XX...XXXXXXXX

or odd (O)

XX...XXX

XX...XXXX

- For the product not to be even what can you say about the
differences?
- What does this mean about the original three numbers?

Now move on to the second part of the problem. Some help with
describing the three types of number related to multiples of three
might be needed (see the notes to the problem

Take
Three from Five ).

### Key questions

- Why are there only three and six differences in the lists?
(because $(a-b)$ is numerically equal to $(b-a) etc$)
- How can we describe all numbers in terms of muliples of $3$, or
$4$ or $5$ ...?
- For the product not to be a multiple of three what can you say
about the differences?

### Possible extension

How many integers do you need to ensure that the product of all
the differences is divisible by $5$?

Abler students can investigate this context more thoroughly,
including posing and pursuing their own questions. For example:
What about divisibility by $4$ and $6$, and then generally ?

Odd
Stones and Take
Three from Five might provide suitable follow-up problems.

### Possible support

Problems like

Ewa's
Eggs and

Make
37 might be good to try first.