Problem
COM-B2-M05-P019 Parity on a Chessboard
On a \(14\times18\) board coloured like a chessboard, some cells are marked. In every row and every column, an odd number of cells is marked. Prove that the number of marked white cells is even.
Use cell coordinates and count modulo \(2\).
Number the rows \(1,\ldots,14\) and the columns \(1,\ldots,18\). Let white cells be those for which \(i+j\) is even. Let \(x_{ij}\) be the indicator of whether the cell is marked.
Modulo \(2\), the number of marked white cells is
\[\sum_{i,j}(1+i+j)x_{ij}.\]
Expanding modulo \(2\), we get
\[\sum_{i,j}x_{ij}+\sum_i i\sum_j x_{ij}+\sum_j j\sum_i x_{ij}.\]
The total number of marked cells has parity \(14\), because each of the \(14\) rows contains an odd number of marks; hence this contribution is \(0\). Row and column mark sums are odd, so the remaining value is
\[\sum_{i=1}^{14}i+\sum_{j=1}^{18}j=105+171=276\equiv0\pmod2.\]
Therefore the number of marked white cells is even.
A stronger version of the previous problem: the colour, size, and formula change.