# How do you find correct eight out of eleven? ### A trip into the eleven-dimensional

A useful clue comes from geometry. However, you have to go into the eleven-dimensional space for this, and that seems a bit confusing, at least on the first visit. A point in this space is nothing more than a sequence of numbers (a “vector”) of length 11. So every attempted solution to our strange exam is such a vector of eleven, with components each of which is either 0 or 1.

In our familiar three-dimensional space, the eight vectors whose components are 0 or 1 form a cube. Accordingly, the conceivable exam solutions are the 2048 corners of the »unit cube in eleven-dimensional space«. If two corners of the ordinary cube differ in only one coordinate, then they are connected by an edge. This should also apply to the eleven-dimensional unit cube. Only here eleven edges go out from each vertex.

In order to find one’s way around in this space better, one introduces something like a distance measurement (“metric”). The distance between two corners of the unit cube is defined as the number of edges that you have to traverse to get from one corner to the other – using the shortest route, of course. It is equal to the number of coordinates in which the two eleven vectors differ. It is also called “Hamming distance” after the mathematician Richard Wesley Hamming (1915–1998), who used this concept of distance to construct error-correcting coding. In more familiar terrain, more specifically in the checkerboard layout of downtown Manhattan, such a distance measurement is known as the “cab driver metric”: the distance from A to B is not the linear distance, but the number of trips from intersection to intersection that you have to cover in the road network.