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This activity has been particularly created for the excepionally mathematically able. (The pupils that you come across in many classrooms just once every few years.)
It can be used as a follow-on fromĀ  Buckets of Thinking..



In this challenge, buckets come in five different sizes: the capacity of a bucket is $2$ litres, $3$ litres, $4$ litres, $5$ litres or $6$ litres. You can choose any number of buckets from two to six, including two and including six. Here are some pictures of buckets. The colours do not matter - they are just to make them look nice!
MultiB

Suppose we choose to use four buckets which each hold $5$ litres, like this:-


4,5 B
We're going to pour water into the buckets, sticking to these rules:

RULE $1$ :- All the buckets must have a different number of litres of water.
RULE $2$ :- Every bucket must contain some water.
RULE $3$ :- Only whole numbers of litres may be used (so no halves, thirds etc.).

So, I'll work this one with you:

In the four buckets you could have:

$1, 2, 3, 4$ litres
or $2, 3, 4, 5$ litres
or $1, 3, 4, 5$ litres
or $1, 2, 3, 5$ litres
or $1, 2, 4, 5$ litres

This looks like all the possibilities obeying the three rules above.
You might like to check that you agree there aren't any other combinations.
Can you explain how you know we've got them all?

Your challenge is to
A/ Choose a number of buckets
B/ Decide on the size they will all be
C/ Find all the different possibilities obeying the three rules above.

You might then like to try again with a different choice.
When you've done that you could compare the two sets of answers and maybe make some suggestions. Let us know what you come up with!