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Fuel Dumps on a Circular Racetrack

Puzzle

A set of fuel dumps on a circular racetrack have just enough gasoline for one car to make one round trip. Prove that there exists a fuel dump from which one car, starting with an empty gas tank, can complete the round trip.

Source

From The Mathematical Gardner (1982), page 283. An article by Ross Honsberger attributes this puzzle to Laszlo Lovasz. Also discussed in Mathematical Puzzles: A Connoisseur's Collection (2003, 163 pages) by Peter Winkler.

Solution

There must exist a fuel dump that allows the car to reach the next fuel dump. Remove the second fuel dump and transfer its fuel to the first fuel dump. Continuing this way, we must be left with one fuel dump. An alternative solution at www.cut-the-knot.org.

Show that a chessboard of size 2^n by 2^n can be tiled with L-shaped figures of 3 squares, such that only one square remains uncovered. In fact, the uncovered square may be any square — for every choice, there exists a tiling. In fact, the puzzle may be extended to 3D: Eight unit cubes make a cube with edge length two. We will call such a cube with one unit cube removed a "piece". A cube with edge length 2^n consists of (2^n)3 unit cubes. Prove that if one unit cube is removed from T, then the remaining solid can be decomposed into pieces.

Next Puzzle: Red Card

Alice repeatedly draws a card randomly, without replacement, from a pack of fifty-two cards. Bob has a one-time privilege to raise his hand just before a card is about to be drawn. Bob must execute his privilege before the last card is drawn. If the card drawn is Red just after Bob raises his hand, Bob wins; otherwise he loses. Is there any way for Bob to be correct more than half the times he plays this game with Alice?

16 Sep 2008
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