Ponder This Challenge - July 2005 - Covering table with coins

Ponder This Challenge:

Puzzle for July 2005. Solutions will not be solicited.

This puzzle was suggested by Alan O'Donnell.
We are not asking for solutions this month.

Upon a rectangular table of finite dimensions L by W, we place n identical, circular coins; some of the coins may be not entirely on the table, and some may overlap. The placement is such that no new coin can be added (with its center on the table) without overlapping one of the old coins. Prove that the entire surface of the table can be covered completely by 4n coins.

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If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

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  Answer:

  Start with the n "small" coins in the rectangle. Replace each one with a "big" coin of twice the radius, at the same center.

  Claim that these completely cover the rectangle; otherwise, a point not covered could serve as the center of a new "small" coin that would not have overlapped with any of the original "small" coins. Now since n "big" coins cover a rectangle of dimension L by W, then n "small" coins will cover a rectangle of dimension L/2 by W/2 (just shrink the whole picture), and four copies of that -- 4n "small" coins -- will cover the original L by W rectangle.

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  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers