Symmetry Arguments
When a puzzle's setup has a symmetry that swaps the two sides, you can often skip the case analysis entirely and read the answer off the symmetry itself.
Prerequisites: How to Attack a Brainteaser
A lot of "who wins the game?" or "can you always split this fairly?" puzzles look like they need a huge case analysis: track every move, every shape, every player's options. Often you don't need any of that. If the setup has a symmetry — a way to relabel or rotate it that leaves the rules unchanged but swaps the two sides — that symmetry itself is usually the whole strategy.
Worked example: the coin-covering game
Two players take turns placing identical circular coins flat on a large rectangular table. Coins may not overlap or hang off the edge. The player who cannot place a coin loses. Who wins, and how?
Try the case analysis first: track coin positions, count remaining space, argue about corners — it gets messy fast. Now try symmetry instead.
The move. The first player places one coin exactly on the centre of the table. From then on, whenever the second player places a coin, the first player places a coin at the point exactly opposite it through the centre — the 180-degree rotation of the opponent's move.
Why it always works. The table is symmetric under 180-degree rotation about its centre, and the centre coin is already placed there, so the remaining free space is symmetric too. Whatever legal spot the second player finds, its mirror image is also free — it cannot already be covered, because the only coin not paired with a mirror partner is the centre one, which the mirror move can't collide with. So the first player always has a legal reply. The second player runs out of moves first.
When a puzzle's rules are unchanged under some relabelling — a rotation, a reflection, a swap of two players — that symmetry is often the entire strategy. Play the one move that creates the symmetry, then mirror everything else.
The same idea answers "can this shape always be cut into two equal-area pieces with one straight line?" for a shape with a centre of symmetry (a rectangle, a parallelogram, an ellipse): any line through the centre splits it into two halves that are 180-degree rotations of each other, hence equal area. Stack two such shapes and the line through both centres bisects both at once — no measuring required.
Ask: is there a point, axis, or relabelling under which the rules stay the same but the roles swap? That is where the symmetry — and the strategy — lives.
Quants lean on the same instinct outside of puzzles. Put-call parity is a symmetry between two routes to the same payoff, and a market maker quoting a two-sided price without a directional view sets bid and ask symmetrically around fair value for exactly this reason: absent new information, there is no basis to break the symmetry.
Related concepts
Practice in interviews
Further reading
- Engel, Problem-Solving Strategies (ch. 1)