Toothpick Puzzles

Geometry is at the core of these puzzles. We challenge you to rearrange these shapes.

Toothpick puzzles (more commonly called “matchstick puzzles”—but it’s safer to use toothpicks than matchsticks) are an old form of geometric puzzle. We’ll try a few of them in this puzzle and try some trickier ones in Toothpick Triangles [1].

Puzzle 1

Remove three toothpicks from the arrangement below to leave exactly three squares.

toothpick_puzzle1

Puzzle 2

Remove six toothpicks from the arrangement below to leave two squares.

toothpick_puzzle2

Puzzle 3

Remove eight toothpicks from the arrangement below to make two squares.

toothpick_puzzle3

Puzzle 4

Remove six toothpicks from the arrangement below to leave three squares.

toothpick_puzzle3

Background

Puzzles like these have been around for more than a hundred years. They help develop geometric visualization skills. If you enjoy puzzles like this, Google “matchstick puzzles” and you’ll find dozens more. Once you get the hang of it, it’s easy to make up your own puzzles.

Course:

Math [2]

Result/Solution(s)

The trick with these puzzles and many more like them is to try to visualize the solution without removing any of the toothpicks. You might think that manipulating the objects themselves would be a good way to go –but surprisingly that’s not the case. As soon as you start actually moving toothpicks you may forget the original arrangement and get confused.

Puzzle 1

You were asked to remove three toothpicks from the arrangement below to leave exactly three squares.

Try to visualize three squares that can be made from these puzzles. There are several, for example, shown with the remaining squares in red.

toothpick_puzzle_1

The first try at visualizing three squares requires five black toothpicks to be removed. The second try, which includes two small 1 x 1 squares and a large 2 x 2 square, requires two black toothpicks to be removed. The third try, on the right, requires exactly three black toothpicks to be removed and therefore solves this problem. This example shows how easy it is to make up problems like this: Start with the same arrangement and change the number of toothpicks to be removed.

Puzzle 2

This time you had to remove six toothpicks from the arrangement below to leave two squares.

tp_puzzle2

The solution to this puzzle leaves two different sized squares after you remove the black toothpicks.

Puzzle 3

This puzzle asked you to remove eight toothpicks from the arrangement below to make two squares.

toothpick_puzzle_3

In this case removing the eight black toothpicks leaves a small 1 x 1 square inside of, but not touching a large 3 x 3 square.

Puzzle 4

Lastly, you were asked to remove six toothpicks to leave three squares.

toothpick_puzzle_4

There are at least two solutions to this one. The first includes a 3 x 3 square with a 2 x 2 square inside it and a 1 x 1 square inside the 2 x 2 square. The second includes a 3 x 3 square with two 1 x 1 squares inside it. You can probably find even more solutions that satisfy this problem.

Math Puzzle [3]