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.
problem
Robot monsters
Age
5 to 7
Challenge level
Use these head, body and leg pieces to make Robot Monsters which are different heights.
problem
6 beads
Age
5 to 7
Challenge level
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?
problem
School fair necklaces
Age
5 to 11
Challenge level
How many possible symmetrical necklaces can you find? How do you know you've found them all?
problem
Reach 100
Age
7 to 11
Challenge level
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.
problem
Factor lines
Age
7 to 14
Challenge level
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
problem
Sealed solution
Age
7 to 11
Challenge level
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?
problem
The dice train
Age
7 to 11
Challenge level
This dice train has been made using specific rules. How many different trains can you make?