Four Square Matchsticks

This matchstick puzzle and many more math challenges can be found in My Best Mathematical and Logic Puzzles by Martin Gardner.

Using four matchsticks of equal length, we can enclose an area of one square matchstick.

Here is an area of 2 square matchsticks enclosed by 6 matchsticks:

Can you enclose an area of 4 square matchsticks using 12 matchsticks?

Safety considerations:

Matches can be dangerous and cause fire. You may use toothpicks or any other sticks of uniform length instead. If you do use matchsticks, do so only with adult supervision.

Course:

Math [1]

Result/Solution(s)

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

6 sqaure matchsticks

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

6 sqaure 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:

4 sqaure matchsticks

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

4 sqaure matchsticks

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.

Math Puzzle [2]