Puzzle

Here is one way to enclose an area of 6 square matchsticks using 12 matchsticks:

To understand how this is true, look at the following figure, in which 12 matchsticks enclose an area of 6 square matchsticks:

The area of this right triangle is half its base times its height, or 1/2 x 4 x 3 = 6.  Another way to look at it is that this triangle is half a 4 x 3 rectangle, whose area would be 12.

By moving 3 of the matchsticks to cut out an area of 2 square matchsticks, the remaining area is 4 square matchsticks.

There are many other ways to solve the puzzle.  Here’s one more. Let’s start with an area of 4 square matchsticks enclosed by 10 matchsticks:

Now let’s remove the two end sticks and substitute a pair of matchsticks for each like this:

The area remains 4 square matchsticks, since the triangular area removed from the left is the same as the one added on the right. But now we have used 12 matchsticks.

There are other patterns like this. Start with 10 matchsticks that enclose an area of 4 square matchsticks. Then find two places where you can substitute 2 sticks for a single stick, adding a triangular area in one place and removing the same area from another place.