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Corners, Edges and Faces Always Come to Two

About 14 minutes

Find a box. Count its corners: eight. Count its edges, the lines where two flat sides meet: twelve. Count its faces, the flat sides themselves: six.

Now take the corners, subtract the edges, add the faces.

8 − 12 + 6 = 2.

Do the same to a dice, a cereal packet, a brick. Eight, twelve, six. You would expect that, because they are all boxes.

So try something that is not a box at all.

Try it on anything you like

A pyramid with a square base has 5 corners, 8 edges and 5 faces. 5 − 8 + 5 = 2.

A tent shape — a prism with triangles at the ends — has 6 corners, 9 edges and 5 faces. 6 − 9 + 5 = 2.

Keep going and it keeps happening:

The solid Corners Edges Faces Corners − edges + faces
A pyramid with a triangle base 4 6 4 2
A pyramid with a square base 5 8 5 2
A tent, or triangular prism 6 9 5 2
A box 8 12 6 2
Two square pyramids glued base to base 6 12 8 2
A six-sided pyramid 7 12 7 2
A pencil, or six-sided prism 12 18 8 2

Look down the last column. Nothing else in the table agrees with itself — corners run from 4 to 12, edges from 6 to 18 — and the answer never moves.

A football is worth doing too, if you have one with panels. It has 60 corners, 90 edges and 32 panels, which is not a small counting job. 60 − 90 + 32 = 2.

Go and find a solid that breaks it. You will not, and the next part is the reason.

Squash the solid flat

Here is the move that makes this obvious rather than mysterious, and it starts by throwing away almost everything about the shape.

Imagine the solid is a balloon with its corners and edges inked onto the surface. Corners and edges do not care about being straight, and faces do not care about being flat. Blow the balloon up and the drawing stretches — the counts do not change, because no corner is created or destroyed by stretching.

So the question is not really about solids. It is about a drawing on a surface: some dots, some lines joining them, and the patches the lines cut the surface into.

Now build such a drawing from nothing, one line at a time.

Three solids drawn as open frames, with a dot at every corner Three different solid shapes side by side, each drawn as an open frame of lines so that the edges at the back show through as well as the ones at the front. On the left a triangular prism with six corners and nine edges. In the middle a cube with eight corners and twelve edges. On the right a pyramid on a square base with five corners and eight edges. A small filled dot marks every corner, so the corners can be counted without touching the shape.

Draw a box onto a balloon from scratch, and watch the total refuse to move.

  1. Start with a single dot on the balloon and nothing else. One corner. No edges. And one face, because with no lines drawn the whole surface is a single unbroken patch. So 1 − 0 + 1 = 2 before you have drawn anything.
  2. Draw a line from that dot to a brand new dot. Corners went up by one, and edges went up by one. They are on opposite sides of the sum, so they cancel exactly. Still 2.
  3. Keep doing that until all eight corners of the box are on the balloon, joined up by seven lines like a branching twig. Corners 8, edges 7, and still just one face, because a twig with no loop in it does not cut the surface anywhere. 8 − 7 + 1 = 2.
  4. Now draw a line between two dots that are already there. This one closes a loop, and a loop cuts one patch into two. Edges went up by one and faces went up by one — and those two are on the same side of the sum, so they cancel too. Still 2.
  5. There are five more lines to draw and every one closes a loop, so every one adds an edge and a face together. Finish the box: 8 corners, 12 edges, 6 faces. 8 − 12 + 6 = 2, and it was never anything else at any point along the way.

Only two things can happen

That is the whole argument, and it is worth stating plainly because it is shorter than it looks.

When you add a line to a drawing, there are exactly two possibilities.

It reaches somewhere new. Then you gained a corner and an edge. In the sum, corners are added and edges are taken away, so the two changes cancel and the total sits still.

It joins two places already on the drawing. Then it must close a loop, and a closed loop divides one patch of surface into two. You gained an edge and a face. Edges are taken away and faces are added, so again the two cancel.

There is no third option. A line has two ends, and each end is either somewhere new or somewhere old.

So the total cannot drift. It starts at 2 with a single lonely dot, and no line you draw is capable of moving it. It is not that every solid happens to come to 2 — it is that no solid is able to come to anything else.

Where it does stop working

A rule this stubborn is worth trying to break, and there is one way to do it.

Take a flat slab and punch a square hole right through the middle, like a picture frame or a washer. Count carefully: 16 corners, 32 edges, 16 faces.

16 − 32 + 16 = 0.

Not 2. And it is not a counting slip — a six-sided washer with a square hole gives 24 corners, 48 edges and 24 faces, which is 0 as well.

The balloon argument quietly assumed something, and the frame breaks it. On a balloon, a closed loop always cuts the surface into two pieces. On a ring, it does not: a loop going round through the hole and back leaves the surface in one piece, so that line adds an edge and no face, and the total drops by one. There are two ways round the hole, so it drops by two.

That is why solids with a hole come to 0 rather than 2. A solid with two holes comes to −2, and the pattern goes on down.

Which means this sum is not really counting corners at all. It is answering the question how many holes does this thing have? — and it will answer it for a shape you have never seen, from three counts made without looking at the shape as a whole.

Six solids. Sort them by what corners − edges + faces comes to. You do not have to count them; you only have to look for the one feature that matters.

  • A box
  • A pyramid on a square base
  • A picture frame: a flat slab with a square hole punched straight through
  • A tent shape with triangles at both ends
  • A washer: a six-sided ring with a square hole through the middle
  • Two square pyramids glued base to base

A solid has 30 corners and 60 edges, and no holes. How many faces does it have?

Go and find the most awkward solid in the house. A crystal, a die with more than six sides, a folded paper shape, a chocolate box with slanted corners. Count its corners, count its edges, count its faces, and watch a number you had no reason to expect turn up again.