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Coin covering puzzle

Posted on: November 27th, 2014 by

100 identical circular coins are placed in a rectangular table such that no extra coin can be placed without overlapping with one of the coins already on the table. For the purpose of this puzzle, assume a coin is placed on a table as long as it's center lies on the table. Show that with 400 coins, one can cover the entire surface area of the table such that there is no patch in the table that is not covered by a coin.
-via AMS puzzle corner

2 Responses to Coin covering puzzle

  1. Sid Hollander had this to say about that:

    Appears problem can be rephrased as the following without distortion.
    4 coins of radius R are placed with their centers on the corners of a square table having a diagonal of 4R. (This is my 'best' worst case'. ie. it ALMOST meets requirements because a coin will fit in the central hole. Now to prove that this square of sides of 2R * root 2 can be painted with 16 coins. As it sits now the square has an area of 8r^2 square units. Utilizing only area of the largesr squares contains a coin it is clear the area may be tiled with the coins available.

  2. Lizard had this to say about that:

    In the original arrangement, every point on the table is within 2R of a coin center (otherwise you could place a new coin there).

    Shrink this arrangement of coin centers by 1/2. Now every point (of the shrunken rectangle) is within R of a coin center, i.e. every point is covered. The original table can be tiled with 4 copies of this.

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