IMO 2025 Problem 6 Visualization
Consider a 2025 × 2025 grid of unit squares. Matlida wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile. Determine the minimum number of tiles Matlida needs to place so that each row and each column of the grid has exactly one unit square that is not covered by any tile.
This visualization helps you explore the solution. The problem asks for the minimum number of rectangular tiles needed to cover an n×n grid so that every row and column has exactly one uncovered unit square. You can explore values of k up to 6 (recall n = k², so n = 36 corresponds to k = 6). For n = 2025, k = 45.
The minimum number of tiles needed for an n×n grid is given by
T(n) = n + 2 floor(√n) − 3.
When n is a perfect square, n = k², this simplifies to T(n) = k² + 2k − 3. The formula is displayed below for the chosen k.
This visualiser focuses on the case where n is a perfect square (choose k using the slider). The structures you see here generalise to larger grids.