Let's Squash a Football

The match is over and everyone has gone in for a drink. A black-and-white football sits alone on the grass. Pick it up and count the black patches. There are 12. Find another ball of the same classic design, from any shop in any country, and count again. Still 12.

That's odd. Footballs come in different sizes. Some are stitched by hand and some are glued by machines. Nobody ever passed a law about black patches. So why does everyone end up with exactly 12?

The answer hides in a tiny rule. It works for a dice, for a camping tent, and for the cut gemstone in a ring. It works for almost any solid shape with flat sides. By the end of this page you'll be able to prove it yourself, using only adding and subtracting. Proving is more than checking. Checking means counting twenty shapes and getting 2 each time. Proving means showing the rule can never fail, on any shape at all. Along the way we'll answer:

• Why does every classic football have exactly 12 black pentagons?
• Why do a dice, a tent and a gemstone all give the same answer?
• Why does a giant dome built from triangles always have 12 special spots?
• Can you build a shape that breaks the rule?
• Why is a donut different from a ball?

We'll start by counting a cube, and add one idea at a time.

1Every solid is made of corners, edges and faces

Think about a closed cardboard box. It has flat cardboard panels. Where two panels meet there is a fold. Where several folds come together there is a sharp point. Run your finger along a fold and it takes you from one sharp point to another.

We'll give these things simple names. Each flat panel is a face. Each fold is an edge. Each sharp point is a corner. (Grown-up books call corners vertices, but corner works fine.)

Below is a cube built from the positions of its 8 corners. Pick what you want to count, then tap each one to mark it. Drag the shape to turn it and find the ones hiding round the back.

A few things to notice:

• On the cube, every corner has exactly 3 edges coming out of it. Mark one corner, switch to edges, and mark its three.
• Switch to the tetrahedron. It has 4 corners, 6 edges and 4 faces. That's the smallest solid you can make from flat faces.
• Try counting the dodecahedron's edges by tapping. It's harder than it looks. Tick see-through to find the ones round the back.
• Pick the football. Its faces are pentagons (5 sides) and hexagons (6 sides). A real leather ball is this shape, pumped up until it's round.

2Count enough shapes and a pattern appears

Counting one shape tells you about one shape. To find a pattern, scientists collect lots of examples and lay them side by side. So let's collect.

Every shape you pick in Sim 1 lands in this table. You can also add shapes with the buttons.

A few things to notice:

• The numbers jump around a lot. The tetrahedron has 4 corners and the football has 60.
• Corners and faces don't seem to follow each other. The cube has more corners than faces. The octahedron has more faces than corners.
• Look along each row. Is there a way to put corners, edges and faces together that gives the same answer for every shape? Try adding two of them, or taking one away from another.
• Stuck? Press Try adding corners and faces. Then compare that yellow column with the edges column in the same row. Do it for every row.

Did you spot it? In every row, corners plus faces is exactly 2 more than edges. The cube has 8 + 6 = 14, and 12 edges. The football has 60 + 32 = 92, and 90 edges. Press Add the last column. It starts with the corners, takes away the edges and adds the faces. The answer is 2, every single time.

If this surprises you, you're in good company. Around 1630 the French thinker René Descartes wrote private notes about solid shapes that contain this pattern in a hidden form. He never published them, and the notes were later lost. Luckily, the German mathematician Gottfried Leibniz had copied some pages by hand in 1676. His copy lay forgotten among his papers until 1860.

The first person to write the rule down plainly was the Swiss mathematician Leonhard Euler. In November 1750 he described it in a letter to his friend Christian Goldbach. He had checked it on every shape he could think of. He also admitted that he couldn't prove it. By section 6, you will have a proof.

3Stretch it or squash it and the counts stay the same

Imagine the cube is made of rubber, like a balloon. You can pull it tall, squash it flat-ish, or blow it up until it's nearly a ball. The faces bend, the edges curve, and the corners move about. But nothing tears, and nothing new gets stuck together.

A few things to notice:

• Pull stretch all the way up. The cube becomes a tall tower block. The counts don't move.
• Turn puff up to the top. The cube turns into a lumpy ball, a bit like a pumped-up football. Still 8, 12 and 6.
• Lumpy pushes each corner in or out by a random amount. Every edge stays joined to the same two corners.

Euler had used this kind of thinking before. In 1736 he cracked a famous puzzle about the city of Königsberg, which had seven bridges over its river. Could you take a walk that crosses every bridge exactly once? Euler saw that the lengths of the streets and the curves of the river didn't matter. All that mattered was which pieces of land were joined by which bridges. (The answer turned out to be no.) It's a cousin of our puzzle.

4Squash any shape flat and nothing changes

Here's a much bolder squash. Take the rubber cube and pop out one whole face, so there's a square hole in the top. Grab the rim of the hole and stretch it wide, wider than the rest of the cube. Then press the whole thing flat onto the table.

You end up with a flat drawing. There's a big square on the outside, a small square in the middle, and four bent squares joining them. Slide to watch it happen.

Where did the popped face go? Look at the shaded space round the drawing. Every bit of that outside used to be the popped face, stretched out enormously. So we count the outside as one face. Then nothing has been added and nothing has been taken away. Same corners, same edges, same faces.

Remember the idea you were holding on to? Stretching and squashing never change what's joined to what. That's why this flat drawing tells the truth about the solid. We'll call it a flat map of the shape.

A few things to notice:

• Stop halfway. The shape looks like a bowl seen from above. Every face is still there.
• Try the football. 31 patches squeeze into the drawing, and the 32nd becomes the outside.
• Try the square pyramid. Its square base becomes the outside, and the four triangles meet at a point in the middle.
• Nothing ever crosses. Each edge still joins the same two corners it always did.

5Snip a fence and two fields become one

Think of the flat map as farmland seen from a helicopter. The edges are fences. The faces are fields. The outside is one more big field that goes on and on. The corners are the posts where fences meet.

Now snip one fence. The two fields on either side become one bigger field. You lost one edge and one face.

What does that do to our total? The total starts with the corners, takes away the edges and adds the faces. Losing a face makes the total 1 smaller. But edges are the part we take away, so losing an edge makes the total 1 bigger. The two changes cancel out. See it with the cube's numbers. Before the snip: 8 − 12 + 6 = 2. After the snip: 8 − 11 + 5 = 2. Taking away 11 instead of 12 gives you one more, and adding 5 instead of 6 gives you one less. Try it.

A few things to notice:

• Snip a fence between two colored fields. They turn into one field. Edges go down by 1 and faces go down by 1.
• Snip a fence on the outer rim. The field inside it joins the outside.
• Keep going until there are no closed fields left. Look at what remains. It's a tree of sticks with no loops at all.
• Try to snip a stick in the middle of the tree. The map won't let you. That would cut the map into two separate pieces.

6Pull off a loose stick and its corner comes too

Once every loop is broken, there are no fences left with a different field on each side. Everything is one big outside field with a tree of sticks lying in it. Look at the tips of the tree. Each tip is a stick with a corner on the end that nothing else touches. The map in section 5 shows these loose ends in red. Scroll back up and tap one of those loose sticks.

The stick goes, and its lonely corner goes with it. You lose 1 edge and 1 corner. Losing a corner makes the total 1 smaller, and losing an edge makes it 1 bigger. They cancel again.

Keep pulling sticks. A tree always has at least two loose ends (try drawing one with no tips at all, and you'll find you've drawn a loop), so you never get stuck. In the end you are left with one single dot sitting in the outside. That's 1 corner, 0 edges and 1 face. 1 − 0 + 1 = 2.

Now run the film backwards.

1. The single dot gives 2.
2. Every snip and every pull kept the total the same. So the flat map gave 2 as well.
3. Squashing the solid flat didn't change any counts. So the solid gave 2.

None of this depended on which shape we started with. Any solid that can be squashed flat this way must give 2. That's the proof Euler couldn't find. In 1813 the French mathematician Augustin-Louis Cauchy published a proof built on this same plan. He flattened the shape, then took it apart one careful step at a time, checking that the total never changed.

Why weren't you allowed to snip a stick in the middle of a tree? That would split the map into two separate pieces. Each piece would end up as its own lonely dot, and two dots in one outside give 2 − 0 + 1 = 3. The rule is about one connected shape, so we keep it in one piece.

7Adding a corner or an edge doesn't change it either

We took a shape apart without changing the total. Building things up works the same way. Here are three ways to add to a flat map, and each one keeps the total the same.

• Put a new corner in the middle of an edge. One edge becomes two. Corners go up 1 and edges go up 1.
• Join two corners with a new fence across a field. One field becomes two. Edges go up 1 and faces go up 1.
• Grow a new stick into a field, with a corner on the end. Edges go up 1 and corners go up 1.

A few things to notice:

• Add a corner to an edge. The total stays the same, even though the drawing now has a corner that isn't pointy at all.
• Join opposite corners of the middle square. It splits into two triangles. One more edge, one more face.
• Try to join two corners with a line that would cross a fence. The map won't allow it, because edges on a solid never cross.
• Press Add one for me twenty times. The map gets crowded, and the total is still 2.

This is why every shape in the table agreed. Any of them can be built from a single dot using these moves, and a single dot gives 2. It also means you can glue extra bits onto a shape, or chop corners off it, and the total still comes out right.

8A football has to have exactly 12 pentagons

Back to the ball on the grass. A classic football is made of two kinds of patches, pentagons and hexagons. And everywhere on the ball, exactly 3 patches meet at each corner. Go back to Sim 1, pick the football, and look closely at any corner to check.

Here's the idea. Every corner and every edge on the ball belongs to the patches round it. If each patch adds up only its own fair share of corners and edges, then all the shares together make up the whole ball, and the whole ball has to come to 2. So let's work out how much one hexagon adds to the total.

A hexagon has 6 corners. But each ball corner is shared by 3 patches, so the hexagon's fair share is 6 ÷ 3 = 2 corners. It has 6 sides, and each edge is shared by 2 patches, so its share is 6 ÷ 2 = 3 edges. And it is 1 face. So one hexagon adds 2 − 3 + 1 = 0. A hexagon adds nothing to the total!

Pentagons are trickier, because 5 doesn't share out evenly into threes. So take six pentagons together. Between them they have 30 corners. Shared by 3, that's 10 ball corners. They have 30 sides. Shared by 2, that's 15 edges. And they are 6 faces. So every six pentagons add 10 − 15 + 6 = 1 to the total.

The total must come out to 2. Hexagons add nothing. Each group of six pentagons adds 1. So you need two groups of six pentagons, and that's 12. The hexagons played no part in it. A ball could have 20 hexagons, or 200, or none at all. It would still need exactly 12 pentagons.

Check it with the calculator. It does the same sums for any number of patches.

A few things to notice:

• Set 12 pentagons and 0 hexagons. That's the dodecahedron from Sim 1.
• Set 12 and 20. That's the classic football: 60 corners, 90 edges, 32 faces.
• Try 11 or 13 pentagons. The corners don't share out evenly, so that ball can't exist.
• Try 6 pentagons, or 18. The sums work out to whole numbers, but the total is 1 or 3. Neither is 2.
• Try 12 and 1. The count says yes, but nobody has ever built that ball, and it turns out nobody can. Passing the count is something every ball must do, but passing it doesn't promise that a ball exists. Why 1 hexagon is impossible is a different puzzle, and it has no one-line answer.

The black-and-white design with 12 pentagons and 20 hexagons became famous at the 1970 football World Cup in Mexico. It showed up well on black-and-white television.

9The same 12 hides in domes, viruses and a molecule

The American designer Buckminster Fuller made geodesic domes famous in the 1950s. A geodesic dome is a ball built from lots of triangles. It's light and very strong. One of the best known is the giant sphere he designed for the 1967 world fair in Montreal.

Look at the corners of a dome. At most of them, 6 triangles meet. But at a few corners only 5 triangles meet. How many? Turn up the size and count the red ones.

A few things to notice:

• At size 1 the dome is just an icosahedron. All 12 corners are red.
• Turn it up. Hundreds of six-triangle corners appear, but the red count stays at exactly 12.
• The red corners sit exactly where the football's pentagons would be, one red corner in the middle of each pentagon.

The same fair-share sums explain it. A corner where 6 triangles meet owns 1 corner, half of its 6 edges (that's 3), and a third of its 6 triangles (that's 2). It adds 1 − 3 + 2 = 0. Six of the five-triangle corners add 6 − 15 + 10 = 1. To reach 2 you need 12 of them, just like the pentagons.

Nature found this out long before Fuller. Many viruses wrap themselves in a protein shell shaped like a dome. The adenovirus, which causes some colds, has a shell built from 252 clusters of protein. 240 of them sit with six neighbours round them. Exactly 12 sit with five.

And in 1985 the chemists Harold Kroto, Robert Curl and Richard Smalley found a molecule made of 60 carbon atoms, shaped exactly like a football. There is a carbon atom at each of the 60 corners and a chemical bond along each of the 90 edges. They named it buckminsterfullerene, after Fuller's domes, and people call it the buckyball. In 1996 the three of them won the Nobel Prize in Chemistry for it. Go back to Sim 1, pick the football and mark every corner. You will have counted the atoms in a buckyball.

10Stranger shapes

So far every shape could be squashed flat like a balloon. What happens with shapes that can't?

A picture frame gives 0

Pick the picture frame. It has 16 corners, 32 edges and 16 faces, and 16 − 32 + 16 = 0. The rule broke!

Think back to the idea you were holding on to. Could you pop one face of the frame and squash it flat like Sim 4? You'd still have a hole in the middle that no amount of stretching can close. A picture frame is a donut in disguise, and stretching can't turn a donut into a ball without tearing it. So a donut is allowed its own number, and that number is 0. Now pick the window with two holes. It gives −2. Every extra tunnel takes another 2 away.

One small detail. The flat top of the frame is a ring, and a face has to be a patch with no hole in it, like a pond. So the model cuts the ring into four pieces. The Swiss mathematician Simon Antoine Jean L'Huilier pointed out shapes with tunnels like this around 1813.

A dent changes nothing

Pick the dented cube. Its top has been pushed in to make a hollow with four triangles. It gives 9 − 16 + 9 = 2. A dent is just another kind of squash. Nothing tore and nothing got glued.

Maps follow the rule too

Draw a round island and split it into 3 countries, like slices of a pizza. Count the countries, plus the sea as one more. That's 4 faces. Count the stretches of border, including the coastline between each pair of border points. There are 3 inland borders and 3 pieces of coast, so 6 edges. Count the points where borders meet. There's 1 in the middle and 3 on the coast, so 4 corners. 4 − 6 + 4 = 2. The flat map in Sim 5 is a map just like this, and the sea is its outside.

Monsters

Pick two cubes touching at a corner. They share a single corner, so you get 15 corners, 24 edges and 12 faces. 15 − 24 + 12 = 3. Now pick the spiky star. Counted as 60 small triangles, it gives 2. Tick count it as 12 big stars. Each colour is now one flat five-pointed star face, and the star gives 12 − 30 + 12 = −6.

For a hundred years after Euler, mathematicians kept finding shapes like these and arguing about them. Some people called them monsters and said they didn't count as proper solids. Others changed the rule to fit them. The philosopher Imre Lakatos told the whole story in a 1976 book called Proofs and Refutations. The fix was to say clearly which shapes the rule is about:

• Every face is a single patch with no holes.
• Every edge touches exactly two faces, and the surface never pinches to a single point.
• Faces don't pass through each other.
• There are no tunnels.

The kissing cubes break the second rule, at their shared corner. The big-star way of counting breaks the third, because the stars pass through each other. A cube with a small cube sitting in the middle of its top face breaks the first. Its top face becomes a ring with a square hole. Build it from toothpicks and clay and count it yourself: you'll get 3, not 2.

11Make your own

A name, at last

You've felt the number now, so it's time for its name. The number you get from corners take away edges add faces is called the Euler characteristic of a shape. Anything you can squash into a ball has Euler characteristic 2. A donut has 0, and a two-hole donut has −2.

The kind of maths that only cares what's joined to what, and lets you stretch and squash as much as you like, is called topology. To a topologist, a cube, a football and a dome are all the same shape. A donut and a coffee mug are the same shape too, because each has exactly one hole.

So the total is always 2 because any ball-shaped solid can be squashed flat and taken apart, one snip at a time, down to a single dot. No snip ever changes the total, and a single dot gives 1 − 0 + 1 = 2.