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Pigeonhole Principle

If you have 12 pigeons and there are only 11 roosts, then at least one roost will be quite cozy.

Level 1

It was around 4 in the morning, and I'm all dressed up, ready for school, when the electricity was cut off. Too bad, I haven't put on my socks yet.

I have 2343 pairs of gray socks, 3212 pairs of pink socks and 6525 pairs of blue socks. Everything is mixed in my drawer (I'm a bit of irresponsible, sorry about that.). As there was no light, I was not able to identify the color of the socks. How many of the socks did I need to take to match one pair?

There are 11 objects that have to have to placed in \(n\) slots. Find the number of maximum possible slots \(n\) such that there are exists at least a single slot in which 3 objects are placed?

Is it true that in any eight composite positive integers not exceeding 360, that at least two are not relatively prime?

A box contains 100 balls of the following colours: 28 red,17 blue, 21 green, 10 white, 12 yellow and 12 black. What is the smallest number \(n\) such that any \(n\) balls drawn from the box will contain at least 15 balls of the same colour?

In a box there are red and blue balls. If you select a handful of them with eyes closed, you have to grab at least \(5\) of them to make sure at least one of them is red and you have to grab at least \(10\) of them to make sure both colors appear among the balls selected. How many balls are there in the box?

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