[Dr. Ashani Dasgupta]

All the squares of a \(2024 \times 2024\)board are coloured white. In one move, Mohit can select one row or column whose every square is white, choose exactly 1000 squares in this row or column, and colour all of them red. Find the maximum number of squares that Mohit can colour red in a finite number of moves.

[Cheenta Support]

Claim : Mohit can colour a maximum of \(3048000=(2024+1024)\times 1000\) unit squares RED

Proof : First, let us prove that colouring \(3048000\) unit squares is possible. Start by colouring the first consecutive \(1000\) unit squares in each of the \(2024\) rows. Then colour the first \(1000\) unit squares of \(1001^{th}\) column to \(2024^{th}\) column. Now this colouring has \((2024+1024)\times 1000=3048000\) unit squares RED.

Now, let us prove that colouring more than \(3048000\) unit squares RED is impossible. Without loss of generality, we can assume that the initial colouring is done to the first consecutive \(1000\) unit squares in the \(1^{st}\) row. This is because square being symmetric, we may consider column as row by just a rotation and we can rearrange the columns to get the first \(1000\) consecutive squares coloured. Now, none of the first \(1000\) columns can be selected for colouring. So, at most \(2024-1000=1024\) column selections can be done and \(2023\) more row selections can be done as there are only \(2024\) rows in total and we cannot select a row (or) column twice. Hence, at most \((2024+1024)\times 1000=3048000\) unit squares can be coloured RED.

• This reply was modified 1 year, 8 months ago by Cheenta Support.

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