Set 12

Globs/Glitches

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Here are two postulates about the two undefined terms globs and glitches.

  1. Every glob has a positive, finite, integral number of glitches.
  2. There are more globs altogether than there are glitches on any one glob.

These two postulates yield a theorem:

There exist two globs having the same number of glitches.

Convince yourself that this theorem is true, and then use it to prove that there are two people in Ohio who have the same number of hairs on their heads. (You’ll have to set it up so that bald people don’t count.)

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Zetters & Zuffers

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The biology classes have completed a lab indicating that there are over a million zetters in a drop of zoop. The zetter is the smallest thing that can be discerned by our microscopes, but it is also known that:

  1. All zetters have zuffers, but none has more than 10;
  2. All zuffers have zeakles, but none has more than 10.

Taking all of these facts into consideration, figure out which of the following statements about a drop of zoop are positively true, and which are possibly false.

  1. Thousands of zetters have exactly the same number of zuffers as thousands of other zetters.
  2. Thousands of zuffers have exactly the same number of zeakles as thousands of other zuffers.
  3. Thousands of zetters have exactly the same number of zuffers, each of which contains exactly the same number of zeakles.
  4. Thousands of zetters contain exactly 5 zuffers.

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Cube, Yellow

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A certain solid cube n inches on a side is made of nice white pine and is painted yellow on the outside. The cube is then cut up into n3 unit cubes; these little fellows have 0, or 1, or 2, or 3 yellow faces. It happens that the number of unit cubes with 1 yellow face is exactly twice the number of those having 2 yellow faces, and there are exactly eight times as many all-white cubes as there are cubes with 3 yellow faces. Find n.

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