mathschallenge.net

Problem

In the 3x3 grid, ⁶/₉ is shaded and in the 4x4 grid, ¹⁰/₁₆ is shaded.

If a 20x20 grid was shaded in the same way, what fraction would be shaded?

Solution

By collecting data for the first few grids and choosing denominators carefully.

It can be seen that ^{(n + 1)}/_2n is shaded.

So in a 20x20 grid, ²¹/₄₀ is shaded.

Can you prove that ^{(n + 1)}/_2n is shaded on an grid?
(Hint: 1 + 2 + 3 + ... + n = ½n(n + 1).)