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An old chestnut goes like this:
 ___ ___
|   |   |
|   |   |
|___|___|
|   |   |
|   |   |
|___|___|
A 2x2 grid of squares are formed by 12 matchsticks as above. Remove two matches to leave just two squares and no extraneous matches that are not part of any square.

Yes, it is really possible to do this! If you think it's impossible, reexamine your assumptions.

There are a number of other closely related puzzles, such as:

 ___ ___ ___
|   |   |   |
|   |   |   |
|___|___|___|
|   |   |   |
|   |   |   |
|___|___|___|
|   |   |   |
|   |   |   |
|___|___|___|
Remove just 4 of the 24 matches to leave only 6 squares.

Answer:

 ___ ___
|       |
|       |
|    ___|
|   |   |
|   |   |
|___|___|
Aye, there's the rub! You have to leave one big square and one small one or you'll never solve this puzzle.

For the second puzzle:

 ___ ___ ___
|   |   |   |
|   |   |   |
|___|___|___|
|   |       |
|   |       |
|___|       |
|   |       |
|   |       |
|___|___ ___|
Note that the two solutions are inconsistent with one another... the former solution counts the big square which has a smaller square contained within it, while this one ignores it and thus counts 6 squares instead of 7.

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