How to get a matching pair of socks?
Suppose there are 24 red socks and 24 blue socks in a drawer. How many socks must I take out (without looking at them) to be sure to get two of the same color?
Solution: Just need to take out three socks. They will either be all the same color or two of one color and one of the other.
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Cutting a rectangle into equal area pieces
In how many ways can you cut a rectangle into pieces of equal area? What if it has a rectangular piece missing in it not in the center. How will you cut that into pieces of equal area?
Solution: There are an infinite number of ways to do this. Any straight cut through the point where the diagonals intersect will do the job.
If there is a rectangular piece missing, a straight cut through the centers of the big rectangle and the small rectangle will do the job.
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How to pay a worker using a single gold bar with two cuts.
You have someone working for you for seven days and a gold bar to pay them. You must pay the worker for the work at the end of every day. Assuming equal pay for each day, if you are only allowed to make two breaks in the gold bar, how do you pay your worker?
Solution: Cut the gold bar into 1/7, 2/7, 4/7 with 2 cuts. Let us call these bars 1, 2 and 4.
At the end of Day 1: Give Bar 1 (You- 2 and 4, Worker- 1); At the end of Day 2: Give Bar 2, Take back Bar 1 (You- 1 and 4, Worker- 2); At the end of Day 3: Give Bar 1 (You- 4, Worker- 1 and 2); At the end of Day 4: Give Bar 4, Take back Bar 1 and Bar 2 (You- 1 and 2, Worker- 4); At the end of Day 5: Give Bar 1 (You- 2, Worker- 1 and 4); At the end of Day 6: Give Bar 2, Take back Bar 1 (You- 1, Worker- 2 and 4); At the end of Day 7: Give Bar 1 (You- Empty, Worker- 1, 2 and 4).