Dots on the table

Question

In ACM posts, August 2008, the “Puzzled” column , Peter Winkler asked the following question:

On the table in front of us are 10 points, and in our pocket 10 $ 1 coins. Prove that the coins can be placed on (without two floors) in such that all points are covered. figure 2 shows the actual placement of coins for this particular set of points; They are transparent, so we can see them. The three coins below are not necessary.

In the next issue, he presented his evidence:

We had to show that any 10 points on the table could be covered with non-overlapping $ 1 coins, in the problem developed by Naoki Inaba and sent to me by his friend Hirokazu Iwasawa, both mavens puzzles in Japan.

The key is to point out that packing discs arranged as a honeycomb pattern are more than 90% of the aircraft. But how do we know what they are doing? A disk with a radius of one fits into a regular hexagon made up of six equilateral triangles of height one. Since each such triangle has an area sqrt(3)/3, the hexagon itself has an area 2*sqrt(3); since hexagons alternate a plane in a honeycomb image, disks, each with an area of ​​π, cover π /(2*sqrt(3))~ .9069 of a flat surface.

It follows that if the disks are placed randomly on a plane, the probability that any particular point is covered is .9069. Therefore, if we randomly place a lot of coins for 1 US dollar (borrowed) on the table in the hexagonal on average, 9,069 of our 10 points will be covered, which means at least some time all 10 will be covered. (We need no more than 10 coins to return the rest.)

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Answer 1

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