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Shading Pattern

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.

Size of Grid  Fraction Shaded
2x2  
3/4
3x3  
6/9 = 2/3 = 4/6
4x4  
10/16 = 5/8
5x5  
15/25 = 3/5 = 6/10

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).)

Problem ID: 101 (Feb 2003)     Difficulty: 1 Star

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