If swapping two objects, reflecting a picture or reversing an order leaves the problem unchanged, then the answer must respect that swap. Two equally likely outcomes each have half of what is left; an integral over a symmetric interval loses its odd part; the best point in a symmetric problem often lies on the line of symmetry.
Symmetry halves work, pairs things up and spots answers before any calculation.
When to try it
Two objects, players or outcomes play exactly the same role.
A diagram has a line or centre of symmetry.
Reversing the order or replacing x by its mirror image gives back the same problem.
Watch out: Check that the symmetry really holds: a condition that treats one object differently (one person must sit at the end) breaks it.
Two worked examples
Try each one first. The hints and the full solution are underneath.
Problem J23
Junior · AlgebraMultiple choiceSolved
Four pencils and three pens cost £2.30. Three pencils and four pens cost £2.60. How much does one pen cost?
Hint
Add the two purchases together, and also find their difference.
Why this works: When two equations are symmetric, their sum and difference are much simpler than the originals. Look for that before reaching for substitution.
Where it leads: Adding and subtracting equations (elimination) generalises to solving any system of linear equations.
How many points with whole-number coordinates lie on the circle x2 + y2 = 65?
Hint
Write 65 as a sum of two squares in every possible way.
Second hint
65 = 1 + 64 = 16 + 49. Count all sign and order variations of (1, 8) and (4, 7).
Full worked solution
Answer: 16
Lattice points on the circle are integer pairs (x, y) with x2 + y2 = 65.
Squares up to 65: 0, 1, 4, 9, 16, 25, 36, 49, 64. Pairs adding to 65: 1 + 64 and 16 + 49 (65 − 0, 65 − 4, 65 − 9, 65 − 25, 65 − 36 are not squares).
So {|x|, |y|} = {1, 8} or {4, 7}.
Each pair gives 2 orders and 4 sign patterns: 8 points, e.g. (1, 8), (8, 1), (−1, 8), ….
Total: 8 + 8 = 16.
Why this works: Lattice points on x2 + y2 = n come from ways to write n as a sum of two squares; symmetry (swaps and signs) multiplies each one by up to 8.
Where it leads: 65 = 5 × 13 is a product of two primes of the form 4k + 1, which is why it has two different representations as a sum of two squares.
A Level Math Revision