By Brittany L. Hayes

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Find all such colorings. Solutions 1. Color the floor as in Fig. 7. A 4 × 1 tile always covers 0 or 2 black squares. A 2 × 2 tile always covers one black square. It follows immediately from this that it is impossible to exchange one tile for a tile of the other kind. Fig. 7 2. Any rectangle with 20 squares can be colored like a chessboard with 10 black and 10 white squares. Four of the tetrominoes will cover 2 black and 2 white squares each. The remaining 2 black and 2 white squares cannot be covered by the T-tetromino.

3 Fig. 4 Fig. 5 12. A beetle sits on each square of a 9 × 9 chessboard. At a signal each beetle crawls diagonally onto a neighboring square. Then it may happen that several beetles will sit on some squares and none on others. Find the minimal possible number of free squares. 2. Coloring Proofs 27 13. Every point of the plane is colored red or blue. Show that there exists a rectangle with vertices of the same color. Generalize. 14. Every space point is colored either red or blue. Show that among the squares with side 1 in this space there is at least one with three red vertices or at least one with four blue vertices.

Then , 1 , 2 , 3 define a tetrahedron T the remaining n − 4 planes cuts T , so that T is one of the parts, defined by the n planes. If the plane would cut the tetrahedron T , then would have to cut at least one of the edges AD, BD, CD in a point Q having an even smaller distance from than D. Contradiction. This is valid for any of the n planes. If there are vertices on both sides of a plane, at least two tetrahedra then must rest on this plane. It remains to be shown that among the n planes there are at most three, so that all vertices lie on the same side of these planes.