The Geometry of Frustration and Discovery

Whole numbers are generally easier to work with than numbers with a lot of digits after the decimal place. The puzzles posed in this post appear to be purely about whole numbers; they ask students to put squares on a grid with a spacing of one. The answers to the puzzles, however, require the use of radicals. The post thus opens the door to irrational numbers. Occupy Math’s editor said that she thinks her personal journey through these puzzles would be increasing frustration ending in rage and a sense of having been deceived when the notion of rotating the square finally occurred to her or was suggested by a collaborator. This perhaps speaks to the need for the teacher or parent to be ready with a hint if the frustration is too deep. It is also worth stating that the reward for the pain of frustration is the sense of accomplishment that comes from understanding or, even better, discovering a new thing.


This post fits into Occupy Math’s series of activity posts. It is easy to learn that the area of a square is the square of the length of its side. In fact we named the mathematical operation “squaring a number” after the area formula for geometric squares. Here is a puzzle problem: Put the corners of a square on the points of a regular grid with points at a spacing of one so that the square has an area of 5. Initially, since the square root of five is not a whole number, this seems like a very hard problem — the trick appears in the picture above. Each side of the tilted square is the hypotenuse of a right triangle with side lengths 2 and 1 — making the side length √5. It is possible to practice and review quite a bit of geometry with puzzles on regular grids with grid spaces at distance one. These puzzles require the Pythagorean theorem and the area formula for rectangles and triangles.

Once you have the idea of tilting a square to get the corners onto a regular grid, there are a large number of possible puzzles. Some examples are given below. All of these are based on the simple rule that the area of the square is the sum of the squares of two whole numbers that are the over-by-up of a right triangle whose hypotenuse is the correct length. For the six examples the two whole numbers are 1;2 2;2, 3;3, 2;4, 2;5, and 4;4. Try making a square of area 17 using these techniques. The number 17 is the sum of which two squared whole numbers?


It may be possible to cause trouble by asking the students, once they discover the tilted square trick, to find a square that need not or even cannot be tilted. Area 16 is an example — 16 is not the sum of two squares unless one of them is zero: 0;4 does the trick but the square is not tilted. These exercises are probably good for grades 4-8, but the real level is determined by having learned the area formula for a rectangle and the Pythagorean theorem. Knowing there are square roots is a prerequisite, which may up the grade level a bit.

More complicated shapes

The next several problems can be solved by remembering that a triangle has an area of one-half base times height. If the triangle is a right triangle, this is just half the product of the length of the legs. One way to see this is to notice that a right triangle is half a rectangle.


Look at the shape above. It breaks into a 6×4 rectangle and a 6×2 right triangle which have areas of 24 and 1/2 of 12 or 24+6=30. You can change the size of the rectangle and triangle to make lots more problems.


This shape breaks into a 4×4 square, a 4×1 triangle, and a 4×2 triangle with areas 16, 2, and 4, totaling an area of 22. In fact, it is possible to go as far as adding a right triangle to all four sides of a rectangle, if you want a fairly hard problem.


The triangle above is inside a 5×5 rectangle. The parts outside of the triangle are right triangles of size 4×2, 3×5 and 1×5. The area of the blue triangle is thus the area of the square minus the areas of the three bounding triangles. The area is 25-1/2×(8+15+5)=25-14=11. As with the other problems, moving the points where the triangle contacts the square (or rectangle) and changing the size of the square (or rectangle) generates many additional problems.

If you are tired of triangles


There are lots of problems on grids that do not use triangles. The blue area above can be computed by just counting the grids, which gives us an area of 48, or we can think of the blue area is an 8×8 square less four 2×2 squares giving an area of 64-4×4=64-16=48. This is an important point — there are often several different ways to solve grid geometry problems. Problems like this one, that have a long solution by counting squares and a short one that requires the student to notice a pattern, encouraging students to use math to avoid work.

Going off the grid

More advanced students can solve for the area of a triangle inscribed into a square or rectangle without help from grids. The students can still figure out the area of the surrounding triangles. Here are the abstract problem statements and solutions. More advanced students, maybe ninth grade or higher, can be asked to derive the general formulas.


Here are the general solutions to these problems:

This activity reviews the area rules for triangle and rectangles. Depending on the puzzle it also gives or encourages practice with pattern recognition and problem solving. The problems that can be solved by counting the grids can be scaled up until counting the grids is tedious, thus strongly encouraging the use of geometric formulas to solve the problem. If you find especially good problems of this sort, Occupy Math would love to hear about them. Speak up in the comments.

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics


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