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Combinatorics Difficulty 4.5 AIME Find the answer Canada

Nate has a grid made of shaded and unshaded 2 cm by 2 cm squares, as shown.

He randomly places a circle with a diameter of 3 cm on the board so that the centre of the circle is at the meeting point of four squares.

What is the probability that he places the disk so that it is touching an equal number of shaded and unshaded squares?

Pick one

Solution

There are 25 possible locations for the disc to be placed.

In the diagram below, each of these locations is marked with a small black disc if it is touching 2 shaded and unshaded squares (an equal number) and a small white disc if it is touching different numbers of shaded and unshaded squares.

[[IMAGE0]]

Therefore, there are 15 locations where the disc is touching an equal number of shaded and unshaded squares.

This means that the desired probability is $\frac{15}{25}$ , which equals $\frac{3}{5}$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.