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More Geometry Puzzles: Counting Hidden Shapes

Updated on November 05, 2014
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TR Smith is a product designer and former teacher who uses math in her work every day.

Geometric shape counting puzzles and brain teasers can help develop your analytical skills and strengthen your working memory. The trick to solving these puzzles is to work out an efficient counting strategy so that you find all the shapes without missing any or counting some twice. Solving puzzles and brain teasers on a regular basis -- whether you prefer pencil and paper games or cell phone apps -- keeps your brain agile and healthy, just as much as eating protein does. These kinds of games and math challenges are also great for elementary school children learning about shapes and geometry for the first time.

Try these four puzzles and check your answers with the solutions that follow. For another set of four geometry puzzles, see companion article.

How many pentagons are in the figure below?

How many rectangles are in the figure below?

How many triangles are in the figure below?

How many circular arcs are in the figure below?

Pentagon Puzzle Solution

A pentagon is any flat shape with five sides, the shape does not necessarily have to be convex. The easiest way to catch all the pentagons in this puzzle is to start by counting all the shapes that are made up of one piece, then two pieces, then three pieces, etc.

Starting this way, you see that there are 2 made up of 2 pieces, 5 made up of 3 pieces, 1 made up of 4 pieces, and 1 made up of 6 pieces. Since 2 + 5 + 1 + 1 = 9, you get a total of 9 pentagons hidden in the diagram.


Rectangle Puzzle Solution

A rectangle is a 4-sided shape with 90-degree angles at every corner. Many people assume that a rectangle must have two pairs of unequal sides, but in fact, a square is also a type of rectangle. Therefore, when you count the rectangles in this puzzle you must also count the squares.

Counting first the square rectangles, there are 5, including the two that are oriented like diamonds. And counting the non-square rectangles there are 9. Therefore, the total number of rectangles is 5 + 9 = 14.


Hexagon Puzzle Solution

The symmetry of this puzzle makes counting the triangles easier. In the hexagon diagram there are 5 types of triangles, labeled A, B, C, D, and E and colored in green in the solution key. Of the Type A triangles there are 6, of Type B there are also 6, of Type C there are 12, of Type D there are 6, and of Type E there are only two.

Therefore, the total number of triangles in the puzzle is 6 + 6 + 12 + 6 + 2 = 32.


This circle has 7 points along its border, and therefore it contains 7*6 = 42 arcs.
This circle has 7 points along its border, and therefore it contains 7*6 = 42 arcs.

Circle Puzzle Solution

This is the hardest puzzle because at first glance there are a lot of distinct arcs. One thing to keep in mind is the definition of a circular arc, which is simply a section of a circle's circumference. Also, it will help you to realize that any arc in the diagram can only be on one circle. Thus, you can count the number of arcs on each of the six circles and add them up.

The number of arcs on any given circle is a function of how many intersection points there are along the circumference. If a circle's boundary is intersected n times, then there are "n choose 2" or (n C 2) ways of picking the endpoints of the arc. But since there are 2 arcs for every pair of endpoints, there are 2*(n C 2) distinct arcs. The expression 2*(n C 2) works out to be n(n-1) or equivalently n^2 - n.

Going from left to right, the first circle has 2 intersection points, the next one has 6, the next has 4, the next has 2, the next has 4, and the last has 2.

The total number of arcs is then

2(2-1) + 6(6-1) + 4(4-1) + 2(2-1) + 4(4-1) + 2(2-1)
= 2*1 + 6*5 + 4*3 + 2*1 + 4*3 + 2*1
= 2 + 30 + 12 + 2 + 12 + 2
= 60


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