, 2018

If we take a $2 \times 100$ (or $100 \times 2$) grid of unit squares, and remove alternate squares from a long side, the remaining $150$ squares form a $100$-comb. Henry takes a $200 \times 200$ grid of unit squares, and chooses $k$ of these squares and colours them so that James is unable to choose $150$ uncoloured squares which form a $100$-comb. What is the smallest possible value of $k$?