Problems

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Problem #PRU-5153
 12–16  4.0

Can you cover a \(13 \times 13\) square using \(2 \times 2\) and \(3 \times 3\) squares?

Problem #PRU-5156
 12–16  4.0

A \((2n - 1) \times (2n - 1)\) board is tiled with pieces of the following possible types:

image

Prove that at least \(4n-1\) of the first type have been used.

Problem #PRU-109487
 12–15  4.0

In an \(n\times n\) table, two opposite corner squares are black and the rest are white. We wish to turn the whole \(n\times n\) table black in two stages. In the first stage, we paint black some of the squares that are white at the moment. In the second stage, we can perform the following two operations as much as we like. The row operation is to swap the colours of all the squares in a particular row. The column operation is to swap the colours of all the squares in a particular column. What is the fewest number of white squares that we can paint in the first stage?

An example of the row operation: let W stand for white and B stand for black and suppose that \(n=5\). Also suppose that a particular row has the colours WWBWB. Then performing the row operation would change this row to BBWBW.