Want a Challenge ?— Try This Overlapping Rectangles Problem

Find the Largest Number of Squares than CANNOT be Covered

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The Australian Intermediate Mathematics Olympiad is a 4 hour competition for exceptional students in Years 7 to 10. This “overlapping rectangles” problem was the final problem in the 2024 competition.

A number of 5 by 7 rectangles are placed on a large grid of unit squares so that each rectangle covers exactly 35 unit squares. The rectangles may be placed horizontally or vertically and they may overlap.

Find the largest integer N for which it is not possible to cover exactly N unit squares.

Video Explanation

Where to Start?

There are probably two main ways to attack this problem. One is to think about which numbers can be covered and the other is to…