Question

In: Advanced Math

You're cleaning up your little nephew's toy room. There are T toys on the floor and...

You're cleaning up your little nephew's toy room. There are T toys on the floor and n empty toy storage boxes. You randomly throw toys into boxes, and when you're done the box with the most toys contains N toys.

(a) What is the smallest that N could be when T=2n+1?

(b) What is the smallest that NN could be when T=kn+1?

(c) Now suppose that the number of toys T satisfies

T< n/(n−1)2.

Prove that when you are done cleaning up, there will be (at least) one pair of boxes that contain the same number of toys.

Hint

Argue the contrapositive by assuming that every box ends up a different number of toys. What is the fewest number of toys you could have started with?

Note:

  • Note you are only being asked to explain part (c) of this activity.
  • You do not need to apply the Pigeonhole Principle directly; instead you can engage in "tipping point" reasoning. See the hint provided in the activity.

Solutions

Expert Solution

Colby Messinger answered 2 years ago
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