7 different colors are used for coloring 30 points on a plane. Prove that, regardless the way you color them, there will always be at least five points colored in the same color.
Parent tip
You can enjoy a fun family time with your children coloring the points in different colors and different ways.
If \(n\) items are put into \(m\) containers, with \(n > m\), then at least one container must contain more than one item.
In our problem, the worst case is to have 4 points of each of the 7 colors, which make 28 points in total. And we have two points left (up to 30 points), that whatever the color we choose, we will have five points in the same color, for sure.
Solution
The answer lies in the Dirichlet's box principle, also known as the Pigeonhole principle, which states that:If \(n\) items are put into \(m\) containers, with \(n > m\), then at least one container must contain more than one item.
In our problem, the worst case is to have 4 points of each of the 7 colors, which make 28 points in total. And we have two points left (up to 30 points), that whatever the color we choose, we will have five points in the same color, for sure.
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