a brouwer theorem problem

Brouwer’s fixed point theorem is a fixed point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x₀ such that f(x₀) = x₀. The simplest form of Brouwer’s theorem is for continuous functions f from a disk D to itself. A more general form is for continuous functions from a convex compact subset K of Euclidean space to itself.

Among hundreds of fixed point theorems,^[1] Brouwer’s is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem or the Borsuk–Ulam theorem.^[2] This gives it a place among the fundamental theorems of topology.^[3] The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry. It appears in unlikely fields such as game theory. In economics, Brouwer’s fixed point theorem and its extension, the Kakutani fixed point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Gerard Debreu and Kenneth Arrow.

Several nice applications of the BFPT exist like this : Take two sheets of paper, one lying directly above the other. If you crumple the top sheet, and place it on top of the other sheet, then Brouwer’s theorem says that there must be at least one point on the top sheet that is directly above the corresponding point on the bottom sheet (fig1)

Fig 1: Brouwer theorem at a piece of paper

Brouwer at a cup of coffee (for more info click here….

The three dimensional case was apparently proposed by Brouwer himself as he drank a cup of coffee, although Henri Poincaré and P. Bohl actually proved parts of the theorem before Brouwer. The consequence of the Brouwer fixed point theorem in three dimensions is that no matter how much you stir a cup of coffee, some point of the liquid will return to its original position. That is, assuming that none of the liquid was spilled (fig2)

Fig 2: Brouwer theorem at a cup of coffee.

And here is a problem from all russian math olympiad 1991 4th round which is related to all above (or so I think):

Fig 3: The flies change each time there position at the the vertices of the cube.

At each vertex of a cube there is a fly. At one moment, each fly moves to another vertex, one fly to each vertex. Show that there exist three flies which form a triangle congruent to the one they formed initially (fig3).