Where Did the Missing Square Go?

What the pieces hold

Start with what you can be sure of. A piece of card covers the same number of squares wherever you put it. So count each one.

• The red triangle is 8 squares across and 3 up. A right triangle is half of the rectangle around it, so it covers half of 8 × 3, which is 12 squares.
• The blue triangle is 5 across and 2 up: half of 5 × 2, 5 squares.
• The yellow piece covers 7 whole squares, and the green piece 8.

So the four pieces cover 12 + 5 + 7 + 8 = 32 squares, before the swap and after it.

Neither picture is the triangle

Now count both pictures. Before the swap the four pieces fill the outline with no hole, so the outline holds 32 squares. After the swap they leave one square empty, so the outline holds 32 + 1 = 33 squares. A true 13 by 5 triangle would hold 32½.

So the first shape is half a square smaller than the triangle, and the second is half a square bigger. Neither of them is the triangle, even though both look like it. The difference between the two shapes, 33 − 32, is exactly the one square of the hole.

In the widget, press Lay a ruler. A straight line runs from the bottom corner to the top corner. Before the swap, the edge of the pieces dips just under it where the red and blue triangles meet. After the swap it bulges just above it. Look closer magnifies that corner six times.

Two different slopes

The long edge of the big “triangle” is made of two pieces: the red triangle’s long side and the blue triangle’s long side. For them to make one straight line, they would have to be equally steep.

How steep a slope is can be measured by how far it goes up for each step across. The red triangle goes 3 up in 8 across. The blue one goes 2 up in 5 across. Which is steeper? Make the steps across the same, like finding a common denominator: in 40 squares across, the red slope would rise 3 × 5 = 15, and the blue slope 2 × 8 = 16.

$$\frac{3}{8} = \frac{15}{40} \qquad\qquad \frac{2}{5} = \frac{16}{40}$$

So the blue edge is a little steeper. When the edge goes red first and then blue, it gets steeper halfway along: it bends inward at the joint, like a valley. When it goes blue first and then red, it gets less steep: it bends outward, like a hill. Figure 17.1 puts the two slopes side by side.

The difference is 1 part in 40: the edge bends by only about 1¼ degrees, far too little for an eye to catch on a picture covered in grid lines and bold outlines.

You meet this idea in Grade 8 under similar triangles: two triangles are similar when one is an enlargement of the other, with all their sides in the same ratio. If the red and blue triangles were similar, their long sides would slope the same way and line up perfectly. They aren’t: 8 is not to 3 as 5 is to 2, since 8 ÷ 3 ≈ 2.67 and 5 ÷ 2 = 2.5.

How thin is the missing square?

Draw both long edges on the same grid, the one that sags and the one that bulges. Between them is a very thin, four-sided shape running from corner to corner: a parallelogram, its opposite sides parallel. That sliver is where the missing square went.

Can it really hold a whole square? Here is a way to find its area by counting, with no measuring at all. Put the sliver in a 13 by 5 box, which holds 65 squares. Around the sliver are six shapes you already know how to count: two triangles of 12, two triangles of 5, and two rectangles 5 by 3, of 15 each. They add up to 64. So the sliver is 65 − 64 = 1 square (Figure 17.2).

The square hasn’t vanished. It has been squeezed into a strip 14 squares long. At its fattest, where the edges bend, the strip is only about 1/8 of a square from top to bottom. If the grid squares were 1 cm wide, the bend in each edge would be about 0.8 mm away from the straight line: less than the width of the pencil line on the board.

A picture is not a proof

The swap seems to prove something impossible: that the same pieces can cover 32 squares or 33. A chain of reasoning that seems to prove something false is called a false proof, and a cut-and-rearrange picture like this one is called a dissection paradox.

Every false proof has a wrong step. Here it is the step nobody said aloud: “Both outlines are the same triangle, because they look the same.” Finding that hidden step is the real skill, called critiquing an argument: not just “that can’t be right” but “this step is where it goes wrong, and here is why”.

A true rule did the finding: cutting a shape into pieces and moving them never changes the total area. The pieces still cover 32 squares, so a hole of 1 square proves the outline has changed.

The chessboard that grows

There is an older and even more famous version. Take an 8 by 8 board, like a chessboard: 64 squares. Cut a strip 3 squares high off the top, and cut it along its diagonal into two triangles. Cut the rest, 8 by 5, into two pieces with four sides. Now lay the four pieces out as a rectangle 13 squares long and 5 high, and it holds 13 × 5 = 65 squares.

Cut the same board at 2 instead, and the rectangle is 14 by 6, holding 84 squares: a gap of 20 squares sits open along the diagonal, and nobody is fooled. Only a few cuts come close.

The numbers that fool the eye

Here are the four cuts the widget asks you to find, and the next one:

Square	Rectangle	Difference
3 × 3 = 9	5 × 2 = 10	1 more
5 × 5 = 25	8 × 3 = 24	1 less
8 × 8 = 64	13 × 5 = 65	1 more
13 × 13 = 169	21 × 8 = 168	1 less
21 × 21 = 441	34 × 13 = 442	1 more

Look at the numbers: 2, 3, 5, 8, 13, 21, 34. Each one is the sum of the two before it: 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, and so on. Starting from 1, 1, they are the Fibonacci numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

In every row of the table, the square’s side is one Fibonacci number, and the rectangle’s sides are the one before it and the one after it. And the two areas always differ by exactly 1, first one way, then the other.

Why always exactly 1?

You can see why without multiplying any big numbers. Compare 5 × 13 with 8 × 8. Write 13 as 8 + 5, and 8 as 5 + 3:

• 5 × 13 = 5 × (8 + 5) = 5 × 8 + 5 × 5
• 8 × 8 = 8 × (5 + 3) = 8 × 5 + 8 × 3

The parts 5 × 8 and 8 × 5 are the same, so the difference between 5 × 13 and 8 × 8 is the difference between 5 × 5 and 8 × 3. But that is the row above, turned round: there, 3 × 8 was 1 less than 5 × 5; here 5 × 5 is 1 more than 3 × 8. So the difference is the row above’s difference with its sign flipped.

The same step works on any row, since every Fibonacci number is the sum of the two before it. So each row’s difference is the previous row’s, flipped: −1, then +1, then −1, forever. The very first row, 1 × 2 against 1 × 1, differs by 1, so every row differs by exactly 1. This is called Cassini’s identity, after the astronomer Giovanni Domenico Cassini, who found it in 1680.

The slopes get closer because the ratio of each Fibonacci number to the one before it settles down: 8 ÷ 5 = 1.6, 13 ÷ 8 = 1.625, 21 ÷ 13 ≈ 1.615, 34 ÷ 21 ≈ 1.619, closer and closer to the golden ratio, about 1.618. A square cut exactly in the golden ratio makes a rectangle with no gap at all, but that cut never falls on a grid line, because the golden ratio isn’t a fraction. With whole squares, one square is always left over.

At Science Playground, the Area Illusion is a different kind of disappearing square, of a kind credited to the Japanese puzzle maker Mitsunobu Matsuyama. Four large, equal pieces fit together into a square with a small square hole in the middle, and a small square tile fits the hole. Arrange the four pieces another way and they make a square again, with no hole at all, and the tile is left over. Where did its space go?

This time nothing bends: every edge is perfectly straight, and both shapes really are squares. The trick is subtler. The cuts run from the middle of each side, tilted a little, so that each piece has two square corners: one on the outside of the big square, and one touching the hole. Turn each piece round so that the corner that touched the hole goes to the outside, and the four pieces close up in the middle (Figure 17.3).

So what has changed? The size of the square. In our version (our own numbers, not the exhibit’s), the first square is 20 cm wide with a hole 2 cm wide: 400 − 4 = 396 cm² of card. The second square is made of the same 396 cm², so it is about 19.9 cm wide, since 19.9 × 19.9 is very nearly 396. It has shrunk by about 1 mm, half of one percent. Shaving 1 mm off two sides of a 20 cm square removes about 2 × 20 × 0.1 = 4 cm²: a whole 2 cm hole, hidden in a change you would need a good ruler to see.

It is no accident that 20 × 20 = 19.9 × 19.9 + 2 × 2 (exactly, with the true width 19.899…). A right triangle is hidden in the cuts: its long side, 20 cm, joins the middles of two opposite sides of the square, and its short sides run along one cut (19.899… cm) and straight across to the parallel cut opposite (2 cm). That is Pythagoras’ theorem at work, and the four pieces around a hole are close cousins of a famous picture proof of it, coming in How Many Ways Can You Prove Pythagoras’ Theorem?

Close in area, not in length

So far the pictures fooled us about area. Here is a twist the other way: two shapes that really do get closer and closer, with the area between them shrinking to nothing, and still something stays different.

A school courtyard is a square 10 m by 10 m. You want to walk from the gate in one corner to the far corner, but you have to walk along the paving lines: east, then north, then east, then north, in equal steps.

With one step you walk 10 m east and 10 m north: 20 m. With two steps, each 5 m east and 5 m north: still 5 + 5 + 5 + 5 = 20 m. With ten steps of 1 m each way, it is still 10 × (1 + 1) = 20 m. However small the steps, all the east parts together add up to 10 m and all the north parts to 10 m, so the path is always 20 m.

But the straight line across, the diagonal, is only about 14.14 m (by Pythagoras: 10² + 10² = 200, and 14.14 × 14.14 ≈ 200). With 40 steps the staircase already looks just like it, and the area between the two is only 100 ÷ 80 = 1.25 m². The path hugs the line, and its length doesn’t change at all.

This one is known as the staircase paradox, and it is a warning for anyone who measures a curve by adding up small pieces: the pieces must lie along the curve, not zigzag round it.

• Is a Pattern That Works Ten Times Always True? Another way that convincing evidence can mislead, and why mathematicians want a proof.
• How Many Ways Can You Prove Pythagoras’ Theorem? Picture proofs that really are proofs, including one with four pieces around a square hole.
• Why Do Sunflowers Count in Fibonacci Numbers? Where the numbers of the 64 = 65 board turn up in plants, and the golden ratio.
• What’s Wrong with This Proof That 1 = 2? (Grades 9–12) False proofs in algebra, and the hidden step that breaks each one.
• How Long Is Nepal’s Border? (Grades 9–12) The staircase’s big brother: why a measured length depends on the size of the steps.
• How Does the Mind-Reading Card Trick Work? (Computing, Going Further, Grades 6–8) Another trick with mathematics hidden inside it.