Juku paints exactly 30 unit squares of a table black. After that, Miku covers up exactly 4 rows and 4 columns. Can Juku ensure by the choice of the squares to be coloured that at least 10 black unit squares are left uncovered?
Solution
Juku can paint 30 unit squares in such a way that each row contains exactly 3 black unit squares and each column contains exactly 2 of them (Fig. 1). When Miku chooses 4 rows and 4 columns, they contain at most black unit squares altogether. Hence at least 10 black unit squares remain uncovered.
Fig. 1
Want a route through all this instead of an archive? The track
puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.