## 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` = ½`n`(`n` + 1).)

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