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PrintSELECTION and TRAINING SESSION
Belarus counting and probability
Problem
Construct a tetramino by attaching two dominoes along their sides such that the midpoint of the longer side of one domino is a corner of other domino. This construction yields two kinds of the tetraminoes with opposite orientations. Let us call them S- and Z-tetraminoes, respectively. 
S-tetraminoes Z-tetraminoes
Assume that a lattice polygon can be tiled with S-tetraminoes. Prove that no matter how we tile using only S- and Z-tetraminoes, we always use even number of Z-tetraminoes.
Assume that a lattice polygon can be tiled with S-tetraminoes. Prove that no matter how we tile using only S- and Z-tetraminoes, we always use even number of Z-tetraminoes.
Solution
Consider the following arrangement of numbers in the cells of the lattice (see Fig.1).
It is evident that the sum of the numbers in any S-tetramino is always zero, so the sum of the numbers in all cells of the polygon from the problem condition equals zero. Fig. 2 Fig. 3
Now we can see that the sum of the numbers in any vertical Z-tetramino (see Fig. 2) is always zero, while the sum of the numbers in any horizontal Z-tetramino (see Fig. 3) is or . Therefore in any tiling of with tetraminoes the number of horizontal Z-tetraminoes should be even (the numbers of tetraminoes with sum equals that of with the sum ). In the same way we can conclude that the number of vertical Z-tetraminoes is also even, which finishes the proof.
It is evident that the sum of the numbers in any S-tetramino is always zero, so the sum of the numbers in all cells of the polygon from the problem condition equals zero. Fig. 2 Fig. 3
Now we can see that the sum of the numbers in any vertical Z-tetramino (see Fig. 2) is always zero, while the sum of the numbers in any horizontal Z-tetramino (see Fig. 3) is or . Therefore in any tiling of with tetraminoes the number of horizontal Z-tetraminoes should be even (the numbers of tetraminoes with sum equals that of with the sum ). In the same way we can conclude that the number of vertical Z-tetraminoes is also even, which finishes the proof.
Techniques
Invariants / monovariantsColoring schemes, extremal arguments