DIY · Card 54 · Mathematics · Puzzles
Four pieces of card make a triangle. Swap two of them round and the very same pieces make the very same triangle, with a square missing. Build it, baffle your friends, then find out where the square went.
01The brief
Same four pieces, same triangle, one square short.
4
pieces: two triangles and two L-shapes
1
square that seems to vanish when you swap them
1953
the year magician Paul Curry devised the puzzle
Arrange four coloured pieces into a triangle 13 squares long and 5 squares tall. Now swap the two triangles round and rearrange the other two pieces. The outline is the same, the pieces are the same, and yet there is now an empty square in the middle.
It is one of the most famous puzzles in mathematics, and one of the best lessons in not trusting your eyes. Make it from squared paper in twenty minutes, and then use a ruler, and a little arithmetic, to catch the trick. Like the paper proof of πr², it is all about cutting shapes up and putting them back together, except that this time, something seems to go wrong.
02Kit
Squared paper and scissors. The ruler is the detective.
Tick off what you have
0 of 5 ready
03Method
Draw every corner exactly on the grid. The trick hides in fractions of a millimetre.
Step 1 of 8 · Minute 0
You need a sheet of 1 cm squared paper, a ruler, a sharp pencil, four coloured pens, scissors, and a sheet of thin card and a glue stick to make the pieces stiff.
The pieces don’t fit the first time.
Why
A triangle or L-shape is drawn off the grid.
Fix
Make sure every corner sits exactly on a grid point, and recut any piece that is off.
The trick is obvious straight away.
Why
Thick outlines or wobbly cuts make the bend easy to see.
Fix
Cut just inside thin lines and use pieces at least 1 cm per square, so the bend is a tiny fraction of the size.
You can’t see the bend at all.
Why
It really is tiny: under a millimetre at 1 cm per square.
Fix
Lay a long steel ruler along the edge, and look for light under it near the corner where red meets blue.
04The trick
The red triangle climbs 3 squares in 8, a slope of 0.375. The blue one climbs 2 in 5, a slope of 0.4. They look identical, but they aren’t, so when you put them end to end the long edge has a slight bend in it. Whether it bends in or out depends on which triangle comes first.
Swap the arrangement and zoom in on the corner where red meets blue:
The close-up shows the corner where red meets blue. It sits at height 3, but a true straight edge from corner to corner would pass at 3.077: the “hypotenuse” dents inwards, and the shape is a hair smaller than a 13 × 5 triangle.
Red slope
3 ⁄ 8 = 0.375
Blue slope
2 ⁄ 5 = 0.400
13 × 5 triangle
32.5 squares
The same sleight of hand works for squares and rectangles. Cut a square whose side is a Fibonacci number into four pieces, lay them out as a rectangle, and a square appears or disappears along the diagonal:
The rectangle seems to gain a square: 65 instead of 64. Along its diagonal the pieces don’t quite meet, leaving a long, thin gap with an area of exactly 1. The Fibonacci numbers 2, 3, 5, 8, 13, 21, 34 … always do this: neighbours multiplied give the square of the middle one, plus or minus one.
05The big idea
Looks are
not proof.
Your eyes say the long side is straight. Arithmetic says the slopes are 3⁄8 and 2⁄5, and those are different numbers. When eyes and arithmetic disagree, arithmetic wins.
That is why mathematicians insist on proofs instead of pictures. A picture can hide a tiny error; a careful calculation can’t. The missing square is a picture that lies, and the calculation that catches it.
06The math
1 · The pieces
12 + 5 + 8 + 7 = 32
Red and blue triangles, yellow and green L-shapes.
2 · A true triangle
½ × 13 × 5 = 32.5
Half of base times height.
3 · Cassini’s identity
Fₙ₋₁ × Fₙ₊₁ − Fₙ² = ±1
For Fibonacci numbers, neighbours multiplied.
The pieces always cover 32 squares. A true triangle would be 32.5. In the first arrangement the corner where red meets blue sits at height 3, just below the true edge at 8 × 5⁄13 ≈ 3.08, so the outline encloses 32. In the second the corner sits at 2, just above the true edge at 5 × 5⁄13 ≈ 1.92, and the outline encloses 33: one square too many, hence the hole.
The sliver between the two outlines is a long, thin quadrilateral with corners at (0, 0), (5, 2), (13, 5) and (8, 3). Its area is exactly 1. That is Cassini’s identity at work: 5 × 13 − 8 × 8 = 1, and for the slopes, 3 × 5 − 2 × 8 = −1.
07Real life
Puzzles, magic and Fibonacci numbers, all playing with how we see.
Get Off the Earth
In 1896 the American puzzle-maker Sam Loyd sold a card with a rotating globe. Turn it one way and you count 13 warriors; turn it the other and there are only 12. The pieces of each figure are shared out differently, the same idea as the missing square.
Lewis Carroll’s 64 = 65
Cut an 8 × 8 chessboard into four pieces and rearrange them as a 5 × 13 rectangle, and 64 squares seem to become 65. The author of Alice in Wonderland was fond of this puzzle, built on the same Fibonacci numbers.
Tangram paradoxes
The puzzle writer Henry Dudeney drew two tangram monks that look identical, except one has a foot and the other doesn’t, though both use exactly the same seven pieces. The answer, again, is a small difference spread around the outline.
Sunflower spirals
Count the spirals of seeds in a sunflower head and you usually find neighbouring Fibonacci numbers, such as 34 one way and 55 the other. The same numbers give the missing square its sizes.
The golden ratio
Divide a Fibonacci number by the one before: 8⁄5 = 1.6, 13⁄8 = 1.625, 21⁄13 ≈ 1.615. The ratios close in on the golden ratio, 1.618…, which is why neighbouring slopes like 2⁄5 and 3⁄8 are so hard to tell apart.
Magic tricks
Paul Curry was a magician as well as a puzzle inventor, best known for a card trick called Out of This World. Like a good trick, his puzzle works by making you look at the obvious thing, the hole, and not the hidden one, the edge.
1953
Paul Curry, an amateur magician in New York City, designed the puzzle in 1953. Martin Gardner, who wrote Scientific American’s famous column on mathematical games, featured it in his 1956 book on mathematics and magic, and it has puzzled readers ever since.
Puzzles where rearranged pieces seem to gain or lose area are centuries older; Lewis Carroll played with one in the 19th century. Curry’s genius was to make the missing area a single, obvious square, so the trick hides in plain sight.
08Is and isn’t
The pieces must change size when you move them.
They don’t: card can’t stretch. What changes is the outline. In one arrangement it sags very slightly; in the other it bulges.
The big shape is a triangle.
It only looks like one. A real triangle has a straight long side; here the edge bends where the red and blue pieces meet.
The missing square is lost in cutting errors.
Even perfectly cut pieces leave the hole. The sliver is a precise mathematical area of exactly one square.
Any numbers would work.
The trick needs two slopes that are very nearly equal and a sliver of exactly one square. Neighbouring Fibonacci numbers deliver both.
09Go further
A
Bigger Fibonaccis
Remake the puzzle with 21 × 8: triangles 13 × 5 and 8 × 3. Is the trick harder or easier to spot?
B
64 = 65
Cut a chessboard-sized 8 × 8 square of squared paper into Lewis Carroll’s four pieces, and make a 5 × 13 rectangle. Where is the extra square?
C
Measure the sliver
At the corner, the edge is out of line by about 0.08 of a square. Draw the puzzle 10 cm per square on a big sheet and measure it.
D
Invent your own
Can you design a missing-square puzzle with different numbers? What has to be true of the two triangles?
E
Fool a friend
Show the first triangle, sweep the pieces together, and rebuild it with the hole. Can they find the trick before you show them the ruler?
10Questions
The big shape is not really a triangle. The red triangle slopes at 3⁄8 and the blue one at 2⁄5, so where they meet, the long edge bends very slightly. In one arrangement it dents inwards, in the other it bulges outwards. The difference between the two is a thin sliver with an area of exactly one square.
According to the puzzle writer Martin Gardner, it was devised in 1953 by Paul Curry, an amateur magician in New York City, so it is often called Curry’s paradox. Puzzles in which rearranged pieces seem to change area are much older, going back to at least the 16th century.
A real 13 × 5 triangle would have an area of 32.5 squares. The four pieces add up to 12 + 5 + 8 + 7 = 32 squares. In the first arrangement they cover 32 squares; in the second the outline encloses 33, so one square must be left empty.
All its lengths, 2, 3, 5, 8 and 13, are Fibonacci numbers. Ratios of neighbouring Fibonacci numbers, like 2⁄5 and 3⁄8, are very close to each other, which hides the bend. And neighbouring Fibonacci numbers multiplied always differ from the square of the one between by exactly 1, which makes the sliver exactly one square.
On 1 cm squared paper, draw and cut out a red right triangle 8 × 3, a blue one 5 × 2, a yellow L-shape of 8 squares and a green L-shape of 7. Arrange them into a 13 × 5 triangle one way, then swap the triangles and rearrange the L-shapes: a hole appears.
Safety · Completely safe. Younger children may need help cutting the corners accurately.
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