IMO Shortlist 2014 problem C4


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Dodao/la: arhiva
May 7, 2017
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Construct a tetromino by attaching 2 \times 1 dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them S- and Z-tetrominoes, respectively.

\setlength{\unitlength}{12pt}
\begin{picture}(7, 4)
  \put(0, 2){\line(1, 0){2}}
  \put(0, 3){\line(1, 0){3}}
  \put(1, 4){\line(1, 0){2}}
  \put(0, 2){\line(0, 1){1}}
  \put(1, 2){\line(0, 1){2}}
  \put(2, 2){\line(0, 1){2}}
  \put(3, 3){\line(0, 1){1}}
  % 
  \put(5, 1){\line(1, 0){1}}
  \put(4, 2){\line(1, 0){2}}
  \put(4, 3){\line(1, 0){2}}
  \put(4, 4){\line(1, 0){1}}
  \put(4, 2){\line(0, 1){2}}
  \put(5, 1){\line(0, 1){3}}
  \put(6, 1){\line(0, 1){2}}
  \put(.5, 0){\text{S-tetrominoes}}
\end{picture}
%
\quad\quad\quad
\begin{picture}(7, 4)
  \put(1, 2){\line(1, 0){2}}
  \put(0, 3){\line(1, 0){3}}
  \put(0, 4){\line(1, 0){2}}
  \put(0, 3){\line(0, 1){1}}
  \put(1, 2){\line(0, 1){2}}
  \put(2, 2){\line(0, 1){2}}
  \put(3, 2){\line(0, 1){1}}
  %
  \put(4, 1){\line(1, 0){1}}
  \put(4, 2){\line(1, 0){2}}
  \put(4, 3){\line(1, 0){2}}
  \put(5, 4){\line(1, 0){1}}
  \put(4, 1){\line(0, 1){2}}
  \put(5, 1){\line(0, 1){3}}
  \put(6, 2){\line(0, 1){2}}
  \put(.5, 0){\text{Z-tetrominoes}}
\end{picture}

Assume that a lattice polygon P can be tiled with S-tetrominoes. Prove that no matter how we tile P using only S- and Z-tetrominoes, we always use an even number of Z-tetrominoes.

(Hungary)

Source: https://www.imo-official.org/problems/IMO2014SL.pdf