Math Puzzles Volume 2: More Riddles And Brain Teasers In Counting, Geometry, Probability And Game Theory by Presh Talwalkar

Author:Presh Talwalkar [Talwalkar, Presh]
Language: eng
Format: epub
Published: 2013-01-08T23:00:00+00:00

Puzzle 15: Fruit Label Stickers

Imagine a perfectly round grapefruit that is overly labeled with 5 fruit stickers.

Prove that the grapefruit can be cut in two equal halves in which one of the halves has 4 of the 5 stickers.

Note: A sticker along the boundary of the cut is counted as being on both halves.

Answer To Puzzle 15: Fruit Label Stickers

Pick any two stickers on the grapefruit. Now make your cut along the great circle that contains these two stickers (a great circle is a circle on the boundary of a sphere with the same radius as the sphere).

These 2 stickers can be counted as being on both of the halves. The remaining 3 stickers have to fit into either of the two halves. The possibilities for (one half, other half) are (3, 0), (2, 1), (1, 2), (0, 3). In each possibility, there is some half that contains at least 2 of these 3 stickers. This half ends up with at least 2 of these stickers, plus the 2 stickers from the boundary cut, for a total of at least 4 of the stickers.

Credit: this is adapted from a 2002 Putnam Exam question.

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