AMC 10 Mastery  

Technique Encyclopedia

Parity

What it is

Parity is reasoning about whether a quantity is even or odd, rather than its exact value. Many quantities that are hard to compute exactly are easy to track mod 2 — and often that's all a problem actually needs.

Signals that suggest using it

  • A problem asks whether something is possible, not what its exact value is.
  • The problem involves sums, differences, or counts where individual terms have a clear even/odd status.
  • Tiling, pairing, or partitioning problems, especially on a checkerboard-colored grid.
  • "Prove that ... is always even/odd" or "show that ... cannot happen."

When it's effective

Parity is fastest and most powerful for proving something is impossible, or for instantly eliminating answer choices that have the wrong parity. It turns what looks like a hard combinatorial question into a one-line argument.

When it's not effective

Parity says nothing about magnitude — it can't tell you a value is too big or too small, only whether it's even or odd. If a problem's answer choices are all the same parity, this tool gives you nothing to work with.

Simple example

A frog on a number line

Problem

A frog starts at 0 and each jump moves it by exactly ±3. After 10 jumps, can the frog be at position 5?

Solution

Each jump changes the position by an odd number. The sum of 10 odd numbers always has the same parity as 10 itself — that is, it's even. So after 10 jumps the frog's position must be even. Since 5 is odd, it's impossible — no case-by-case search of jump sequences was needed.

AMC-style example

Toggling switches by divisor count

Problem

A row of 15 lightbulbs, numbered 1 to 15, all start off. For each k from 1 to 15, you toggle the switch of every bulb whose number is a multiple of k. After all 15 rounds, how many bulbs are on?

Solution

Bulb n gets toggled once for every divisor of n, so its final state is on exactly when n has an odd number of divisors. Divisors pair up as d and n/d — unless d=n/d, which only happens when n is a perfect square. So exactly the perfect squares end up on: 1, 4, and 9. 3 bulbs are on.

Invariants Casework Testing Small Cases

Practice