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Puzzled: Solutions and Sources

It's amazing how little we know about good old plane geometry. Last month (August 2010, p. 128) we posted a trio of brainteasers, including one as yet unsolved, concerning figures on a plane. Here, we offer solutions to two of them.
  1. 1. Covering a Gravy Stain.
  2. 2. Covering Dots on a Table.
  3. 3. Placing Coins.
  4. Author
  5. Footnotes

Solution. The object was to cover a gravy stain of area less than one square inch with a plastic sheet containing a grid of side one inch in such a way that no intersection point of the grid fell on the stain. This puzzle (as I was reminded by Andrei Furtuna, a Dartmouth computer science graduate student) may date to the great Lithuanian-born mathematician Hermann Minkowski (1864–1909).

It suffices to consider only grids oriented North-South-East-West or, equivalently, to assume the plastic sheet is aligned with the table. Now imagine cutting the tablecloth into one-inch squares in an aligned grid pattern, pin one of the stained squares to the table, oriented as it was originally, and stack (without rotating any square) all other stained squares neatly on top of it.

The stain is now within one square, but since the area of the stain is less than one square inch (and can be reduced only through stacking), some squares remain stain-free. Now pick a stain-free point and place the plastic so its intersection points lie directly on the point.

Since all other intersection points are outside the stained square, no intersection point touches the stacked stain. But what if the tablecloth were sewn back together? Each stained square would then be translated by an integral number of tablecloth squares East or West and North or South back to the original position. It would then bear the same relationship to the plastic sheet’s grid points it did before; that is, it would miss them.

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2. Covering Dots on a Table.

Solution. We had to show that any 10 dots on a table can be covered by non-overlapping $1 coins, in a problem devised by Naoki Inaba and sent to me by his friend, Hirokazu Iwasawa, both puzzle mavens in Japan.

The key is to note that packing disks arranged in a honeycomb pattern cover more than 90% of the plane. But how do we know they do? A disk of radius one fits inside a regular hexagon made up of six equilateral triangles of altitude one. Since each such triangle has area radic.gif 3/3, the hexagon itself has area 2 radic.gif 3; since the hexagons tile the plane in a honeycomb pattern, the disks, each with area π, cover π /(2 radic.gif 3) ~ .9069 of the plane’s surface.

It follows that if the disks are placed randomly on the plane, the probability that any particular point is covered is .9069. Therefore, if we randomly place lots of $1 coins (borrowed) on the table in a hexagonal pattern, on average, 9.069 of our 10 points will be covered, meaning at least some of the time all 10 will be covered. (We need at most only 10 coins so give back the rest.)

What does it mean that the disks cover 90.69% of the infinite plane? The easiest way to answer is to say, perhaps, that the percentage of any large square covered by the disks approaches this value as the square expands. What is “random” about the placement of the disks? One way to think it through is to fix any packing and any disk within it, then pick a point uniformly at random from the honeycomb hexagon containing the disk and move the disk so its center is at the chosen point.

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3. Placing Coins.

Unsolved. The solution to Puzzle 2 doesn’t tell us how to place the coins, only that there is a way to do it. Is there a constructive proof? Yes, and we can use the solution to Puzzle 1 (concerning the stain) to find it. I leave it to your imagination to follow up.

That proof can be used to increase the number of dots to 11 or 12, still using only an aligned hexagonal lattice of coins. However, since we aren’t restricted to a lattice, it seems plausible that quite a few dots can be covered, perhaps as many as 25 (see the August column). If you figure out a dot pattern with, say, 30 or fewer points you think can’t be covered by unit disks, please send to me, along with your reasoning.

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