Three shape puzzles

Toy 1 · Colours, distance 1

Paint the plane, one rule

Give every point on an endless flat sheet a colour. The one rule: two points exactly 1 apart must never match. How few colours can you get away with? Start small.

Three colours fail on seven points

These seven points sit so that every line drawn is exactly 1 long. Click a point to change its colour. Then let the computer try every way to colour them.

With three colours there are 3⁷ = 2,187 ways to colour the seven points, and every one puts two matching colours at the ends of some line. So the plane needs at least four colours.

Seven colours always work

Tile the floor with hexagons a bit less than 1 across, coloured so a hexagon’s colour only repeats far away. A stick of length 1 can’t have both ends in one hexagon, and it can’t reach the next hexagon of the same colour. Drag the stick around. Then change the tile size and watch it break.

What the model leaves out: the computer only checks seven points here, and the sticks are dropped on a finite patch of floor. The bound of five needs a much bigger graph and a computer search we didn’t rerun.

Toy 2 · One tile for every shape

One tile fits all

Take any flat shape whose widest stretch is 1: no two of its points are more than 1 apart. Is there one tile, as small as possible, that every such shape fits inside after you turn and slide it? Henri Lebesgue asked in 1914. Here is the classic answer: a regular hexagon exactly 1 across between opposite sides.

The computer turns the shape through 60° in small steps and, at each turn, works out whether some slide keeps it inside all six walls. The hexagon’s three pairs of walls are each exactly 1 apart, and a shape 1 across is never wider than 1 in any direction, so one turn always works. The disk that is 3% too wide is the control: it is wider than 1 in every direction, so no turn can work, and the search says so.

What the model leaves out: curved shapes are drawn with a few hundred straight sides, and a fit counts when the shape is inside to the computer’s rounding. Only the plain hexagon is searched here; the trimmed one is drawn, with its area worked out from the cut lines.

Toy 3 · A hallway, a corner, a couch

The sofa and the corner

A hallway 1 wide turns a right-angle corner. What’s the biggest flat sofa you can slide and turn around it? Leo Moser asked in 1966. A unit square works (area 1). A half-disk of radius 1 turns on the spot (area 1.571). You can do better by carving.

Start with a long block. Turn the hallway around it a little at a time, while its inner corner swings along a half-circle, and cut away everything the walls cross. What’s left can make the turn by construction. The slider sets how far apart the two rounded ends sit. John Hammersley worked out this family in 1968: the best split gives area 2.2074. The uncarved block is the control; drive it and watch the walls bite.

What the model leaves out: the sofa is carved on a grid of squares 1/150 wide and the hallway turns in 180 small steps, so the counted area runs slightly above the exact formula. The sofa here follows Hammersley’s path; Gerver’s better shape uses a different path that we didn’t rebuild.

Prior art

Who got there first

The code behind every number runs in your browser: plane.js (the spindle check and the hexagon floor), cover.js (the fitting search and Pál’s areas), sofa.js (the carving and the drive).

Next click down

The sofa is the closest of the three to finished: if Baek’s argument holds up, the answer is Gerver’s 2.2195. The plane is the furthest: going from “at least 5” to “at least 6” would need a new idea, and the search for small five-colour graphs is still a good place to look.