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. Covering a Gravy stain. 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 Lithua-nian-born mathematician Hermann
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
Since all other intersection points
are outside the stained square, no in-
tersection 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 plas-
tic sheet’s grid points it did before; that
is, it would miss them.
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 Iwa-sawa, 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 √3/3,
the hexagon itself has area 2√ 3; since
the hexagons tile the plane in a honeycomb pattern, the disks, each with area
π, cover π /( 2√ 3) .9069 of the plane’s
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 con-
taining the disk and move the disk so
its center is at the chosen point.
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.
All readers are encouraged to submit prospective
puzzles for future columns to email@example.com.
Peter Winkler ( firstname.lastname@example.org) is professor of
Mathematics and of computer Science and Albert bradley
third century professor in the Sciences at dartmouth
college, hanover, nh.