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This essential trains for: AMC-8, AMC-10, AMC-12.

An invariant is a quantity that remains unchanged when an operation is performed.

Problems based on invariants involve an operation or a procedure. The strategy for solving them is based on finding what quantity is not going to change as the operation or procedure is performed repeatedly.

Example 1: The perimeter of a rectangle is invariant with respect to the operation of 'cutting a rectangular corner out.'

Example 2: In a bag, there are 3 red marbles and 3 blue marbles. On the side, you have additional unlimited amounts of red and blue marbles. An operation is defined as follows:

- blindfolded, extract two marbles;
- if the two marbles are red, replace them with blue marbles;
- if the two marbles are blue, replace them with red marbles;
- if the two marbles are of different colors, put them back in the bag;

After how many operations could there be only marbles of one color in the bag?

The invariant is the parity of the number of blue or red beads. It is easy to see that, the number of blue marbles can only change from 3 to 1, 3 or 5. Regardless of the number of operations, the number of blue marbles will remain odd. Similarly, the number of red marbles will remain odd. Since six is an even number, there is no sequence of operations that will produce an even number of marbles of some color.

Problems based on invariants may come from the realms of geometry, arithmetic, counting, algebra - virtually any branch of mathematics. The most creative - and difficult - step is to figure out what the invariant quantity is.