An invariant is a quantity that never changes, however the moves in a puzzle are made: the parity of a sum, a total, a product, or a colouring count. If the start and the target have different values of the invariant, the target can never be reached, and no amount of trying will get there.
A monovariant is a quantity that only ever moves one way (it always increases, say). It proves that a process must stop.
When to try it
The problem describes repeated moves and asks whether a position can ever be reached.
Many choices are possible at each step, so trying them all is hopeless.
Something about the position looks ‘protected’: the total of the numbers, the number of heads modulo 2, the colours of the squares covered.
Watch out: An invariant can prove that something is impossible, but never that it is possible. For that you need an explicit sequence of moves.
Two worked examples
Try each one first. The hints and the full solution are underneath.
Problem J25
Junior · AlgebraShort answerSolved
Jo is three times as old as Sam. In 12 years’ time Jo will be twice as old as Sam. What is the sum of their ages now?
Hint
Let Sam be s years old now. Write Jo’s age now and both ages in 12 years.
Second hint
3s + 12 = 2(s + 12).
Full worked solution
Answer: 48
Let Sam be s years old now; then Jo is 3s.
In 12 years Sam is s + 12 and Jo is 3s + 12.
Then Jo is twice Sam’s age: 3s + 12 = 2(s + 12) = 2s + 24.
So s = 12, and Jo is 36.
Check: in 12 years they are 24 and 48, and 48 = 2 × 24. ✓ Sum now: 12 + 36 = 48.
Why this works: Age problems become easy once every age is written in terms of one letter at one time, then shifted by the same number of years.
Where it leads: The age difference never changes, so at the time Jo is twice Sam’s age, Sam’s age equals the difference: a quicker route.
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