The Same Twelve Squares, a Longer Edge

Cut twelve squares out of paper, all the same size. Push them together into a block four across and three down. Now run your finger round the outside and count the square edges you travel along.

Fourteen.

Break it up and lay the same twelve squares out in a single straight line. Run your finger round that.

Twenty-six.

Nothing was added and nothing was taken away. The floor covered is identical — twelve squares either way — and the line round the outside nearly doubled. That gap is not an accident, and there is a rule that predicts it before you count anything.

The two things worth counting

There are two separate measurements here, and mixing them up is the whole difficulty.

The area is how much floor the shape covers. That is just how many squares you have, and rearranging them cannot change it. Twelve squares cover twelve squares of floor, always.

The edge is the line round the outside. That is what a fence would have to be, or a skirting board, or the wrapping on a parcel. And it changes enormously depending on how the squares are placed.

Real things care about both, separately. A room has a floor to carpet and walls to paper, and rearranging the walls changes one without touching the other.

The rule

Here is the piece that turns counting into predicting.

Take one square on its own. It has four edges, all of them on the outside.

Now push a second square against it. The two squares are the same as ever, but the edge where they meet has vanished from the outside — and two edges vanished, not one, because each square gave up an edge to make the join.

That is all that ever happens. So:

The outside edge is four times the number of squares, minus two for every place where two squares touch.

Try it on the block of twelve. Count the joins: three joins along each of the three rows makes nine, and two joins down each of the four columns makes eight. Seventeen joins in all.

Four times twelve is 48. Take away two for each of the seventeen joins, which is 34. You get 14, which is exactly what your finger found.

The surprise in the middle

Lay the twelve squares out as a staircase, one step at a time, climbing to the right. It looks nothing like a straight line — it turns a corner every square or two, and it sprawls across the page.

Count its edge.

Twenty-six. Exactly the same as the straight line.

That is worth sitting with, because it is the opposite of what the shapes look like. The staircase and the line are not alike at all, but they share a property that decides the answer: neither of them ever has three squares packed round a corner. Every square in both shapes is joined to the next one and to nothing else.

Both shapes have eleven joins, and eleven joins is eleven joins whichever way they are pointing.

The two limits

Because the edge is fixed by the joins, you can work out the best and worst arrangements without trying any of them.

The longest edge. To make the edge as long as possible you want as few joins as possible. But the shape has to hold together, and to hold twelve squares together you need at least eleven joins — one fewer than the number of squares, exactly like eleven links holding twelve beads in a chain. So the fewest joins is 11, and the longest possible edge is 48 − 22 = 26.

That is why the straight line and the staircase tie. They are not merely long; they are both as long as anything with twelve squares can be. Any shape that never closes a loop and never packs squares round a corner will measure 26.

The shortest edge. For the shortest edge you want as many joins as you can pack in, and squares packed into a tidy rectangle share the most. Seventeen is the most you can manage with twelve, so the shortest possible edge is 48 − 34 = 14.

So twelve squares can have any even edge from 14 to 26, and nothing outside that. The 3 × 4 block wins because 12 splits into 3 × 4, which is the nearest twelve gets to a square.

Two edges at once

That ring in the sorter is the odd one out, and it is worth a second look.

A four-by-four square with the middle four squares taken away leaves twelve squares in a ring with a hole. Its outside is a four-by-four square, which measures 16. But it has an inside as well — the hole is a two-by-two square, which measures 8.

Total edge: 24. And the rule agrees. The ring has twelve joins, so 48 − 24 = 24.

The rule never asked whether an edge was on the outside or round a hole. It only ever counted joins, and it gets holes right without being told about them.

Try to build twelve squares into a shape with an edge of 13, and you will fail. Try for 27 and you will fail. Then try the same rule with twenty squares, or a hundred, and watch it hold — because it was never about twelve. It was about what happens every time two squares meet.