# Geometry Brain Teasers: Shape Counting Puzzles

Among mathematical brain teasers, shape counting puzzles are among the simplest to understand, but often the hardest to work out. They rely on visual acuity and memory as much as math and geometry skills, and therefore offer a good, well-rounded brain workout. They're also a great tool for teaching and reinforcing geometry concepts for elementary school kids.

"Count the number of shapes" puzzles are usually two-dimensional line diagrams composed of regular and irregular shapes. Your task is to find all the triangles, squares, rectangles, or other polygons hidden in the figure. For most types of these brain teasers, there are methodical ways you can go about counting all the shapes so that you don't miss any or count any twice. Test your puzzle solving skills with these four geometric brain teasers. Solutions are given at the end.

## How many triangles are in the figure below?

## How many squares are in the figure below?

## How many triangles are in the figure below?

## How many simple polygons are in the figure below?

## Pentagon Puzzle Solution

To count the number of triangles in the pentagon, one approach is to color in the smallest self-contained pieces and then count the number of triangles according to how many pieces they contain.

In the figure, there is 1 triangle made up of 5 pieces, 2 triangles made up of 4 pieces, 4 triangles made up of 3 pieces, 10 triangles made up of 2 pieces, and 9 triangles made up of 1 piece.

The total number of triangles is 1 + 2 + 4 + 10 + 9 = 26.

## Square Puzzle Solution

The square puzzle is tricky because there are many squares composed of smaller square and rectangular pieces, and there is another square overlaying the 3x3 grid. To count all the squares, first ignore the area shaded in green and consider only the 3x3 grid. In the 3x3 grid, there are 9 small 1x1 squares, 4 medium 2x2 squares, and 1 large 3x3 squares, for a subtotal of 14.

Now considering all the squares that can be formed from parts of the green region, there are 4 small squares at the corners, 4 larger squares at the corners, and 1 large square whose boundary is the green region, for a subtotal of 9 additions squares.

Therefore, the total number is 14 + 9 = 23 squares.

## Triangle Puzzle Solution

Like the pentagon puzzle, this triangle counting puzzle can be solved by coloring all of the self-contained pieces a different color and counting the number of triangles that can be formed formed 1, 2, 3,... 9 different pieces.

In this puzzle, there is 1 triangle that can be formed from all 9 pieces, 0 triangles that can be formed with 7 or 8 pieces, 2 triangles that can be formed from 6 pieces, 2 triangles made up of 5 pieces, 3 triangles made up of 4 pieces, 7 triangles composed of 3 pieces, 6 triangles made of two pieces, and 7 triangle consisting of one piece.

The total number of triangles is 1 + 0 + 0 + 2 + 2 + 3 + 7 + 6 + 7 = 28.

## Polygon Puzzle Solution

The key to counting the number of simple polygons in the last puzzle is knowing the definition of a simple polygon in plane geometry. A simple polygon is any two-dimensional connected figure whose boundary is formed by straight lines, whose boundary lines do not self-intersect. They can be convex or concave. If you only count the convex polygons, you will miss most of them. Counting all the convex and concave polygons in the figure, you should find 23.

## How did you do?

Don't worry if you don't find all the shapes the first time around. The more frequently you do puzzles and play games, the better you get. This goes for all kinds of games and puzzles, be they crosswords, mazes, phone games, video games, etc. The brain is like a muscle and exercising it consistently makes it stronger and healthier.