An invariant is something that stays the same while everything else seems to move. Invariant thinking is one of the cleanest ways to prove that a task is impossible.
Many math problems describe a process: flip coins, move tokens, rearrange tiles, change signs, combine numbers. The problem asks whether a certain final state can be reached. Trying every possible sequence is usually hopeless. Instead, we look for a quantity that never changes.
If the starting state and ending state have different values of that quantity, the ending state is impossible.
Parity as the first invariant
Parity means whether a number is even or odd. It is often the first invariant students can use.
Suppose a game begins with an even number of black tokens. Each move changes two tokens from black to white or two from white to black. Can the game end with an odd number of black tokens?
No. Every move changes the number of black tokens by 2, -2, or 0. The parity of the black-token count never changes. Starting even means always even.
This proof is short, but it models a deep move: do not track everything. Track the right thing.
Coloring arguments
Chessboard coloring creates many invariant arguments. A domino covers one black square and one white square on a normal checkerboard. If two opposite corners are removed from an 8-by-8 board, both removed squares are the same color. The remaining board has unequal numbers of black and white squares.
Because every domino covers one of each color, the board cannot be tiled by dominoes.
This argument avoids trying tile placements. It uses a quantity each tile must preserve: the difference between black and white covered squares.
Students often find this satisfying because the impossibility becomes visible. The board looks almost complete, but the coloring reveals what is wrong.
Invariants in games
Consider a game where two players remove stones from piles according to certain rules. Sometimes the winning strategy depends on a quantity that the current player can force to remain favorable.
For a simple example, imagine a pile of 21 stones. Each player removes 1, 2, or 3 stones. The player who takes the last stone wins. The key positions are multiples of 4. If you leave your opponent 20, 16, 12, 8, or 4 stones, then whatever they remove, you remove enough to return to the next lower multiple of 4.
Here the invariant is not exactly constant after every single move. Instead, the strategy repeatedly restores a controlled condition. Students should see that invariant-style thinking can appear as maintaining a structure, not merely watching a number stay fixed.
Monovariants
A monovariant is a quantity that always moves in one direction. It might always increase or always decrease. Monovariants are useful for proving a process must stop.
If every move decreases a positive integer score, the process cannot continue forever. It must end because there are only finitely many positive integers below the starting score.
This kind of reasoning appears in algorithms. To prove an algorithm terminates, we often define a measure of progress that decreases every step.
How to search for invariants
Students can ask these questions:
- Does parity stay fixed?
- Does a remainder modulo some number stay fixed?
- Does a coloring balance stay fixed?
- Does a total, difference, or product stay fixed?
- Does some quantity always increase or decrease?
The first attempted invariant may fail. That is normal. A failed invariant is not wasted work; it teaches us what the moves actually change.
A COSMO teaching note
Do not introduce invariants by giving the definition first. Start with an impossible-looking process. Let students try moves. Ask what they notice. Then name the stable quantity after they have felt the need for it.
The word invariant should arrive as a label for a tool students already used.
The core idea
Invariant thinking is a way of refusing to be distracted by motion. A process can look chaotic, but if one carefully chosen quantity cannot change, then some destinations are unreachable.
In problem solving, that is a powerful sentence: no matter what you do, this part stays the same.