Things that cannot be done — page 11
Algebra
Three coordinates and a fourth power
The Heisenberg group is made of triples of whole numbers, and two moves generate all of it. Count the triples within r moves of the start and the count grows like r⁴, not r³: at forty-four moves there are 1,600,703 of them, against 117,569 for the cubic lattice. The fourth power comes from one coordinate that the moves reach by enclosing area, so that height costs only the square root of itself — and seen from far away, the ball of reachable triples is a curved solid with a dimple at each pole.
Applied
A party that loses every issue can win
Two parties take opposite sides on three issues, and every voter backs the party they agree with on more of them. The party that wins the election can be on the losing side of all three issues — in a fifth of random electorates it loses most of them, and in about one in a hundred it loses every one. A majority of voters can even find themselves outvoted on most of the questions decided. Only when issues are settled by three-to-one majorities on average is that ruled out.
Discrete
No triangle, and four colours
A graph that contains four mutually joined points needs four colours. The converse looks obvious — a graph needing four colours should contain something like that — and it is false. Grötzsch's graph has eleven points, no three of them mutually joined, and needs four colours. Mycielski found a step that adds one colour to any graph without creating a triangle, so graphs with no triangle can need any number of colours, and their need is spread through the whole graph, with no part to point at.
Geometry
A cube through a hole in a cube
Prince Rupert of the Rhine bet that a hole could be cut through a cube large enough for an equal cube to pass through it, and won. The trick is to look at the cube along a direction near its long diagonal, where its shadow is a hexagon wide enough to hold a square face with room to spare: a cube 6% larger can pass. All five Platonic solids turn out to have such a passage, found by searching over the directions from which a solid is seen — and in 2025 a solid was found that has none.
Logic
No way to hand out true and false
Birkhoff and von Neumann read the propositions of quantum mechanics as subspaces, with "and" the intersection and "or" the span. Three lines in a plane already break the distributive law. Kochen and Specker proved something stronger: the propositions cannot be called true and false at all, if every complete measurement is to have exactly one true outcome. Cabello's eighteen directions in four dimensions show why in one line of arithmetic — nine bases, each direction in two of them, and nine is odd — and a search of all 262,144 assignments confirms it.
Probability
Two losing games that win together
Game A is a coin that wins 49.5% of the time. Game B tosses a bad coin when the capital is a multiple of three and a good one otherwise, and it loses too, because the capital spends more than a third of its time on multiples of three. Choose between the two games at random and the walk drifts upwards by 0.0157 a round; play A, B, B over and over and it gains 0.0574. Nothing is wrong with the arithmetic. The losing coin A wins by knocking the capital off the bad remainder.