Proof by exhaustion depends on there being a small number of results so that it is manageable to find all possibilities; it is about working systematically. The tasks below all offer opportunities to use proof by exhaustion. They form part of our Mastering Mathematics: Developing Generalising and Proof Feature.

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### Robot Monsters

Use these head, body and leg pieces to make Robot Monsters which are different heights.

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### 6 Beads

If you put three beads onto a tens/ones abacus you can make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?

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### School fair necklaces

How many possible symmetrical necklaces can you find? How do you know you've found them all?

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### Reach 100

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

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### Factor Lines

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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### Sealed Solution

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

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### The Dice Train

This dice train has been made using specific rules. How many different trains can you make?