solutions and sources
Last month (May 2010, p. 120) we posted a trio of brainteasers, including
one as yet unsolved, concerning variations on the Ham Sandwich Theorem.
The Intermediate Value Theorem says that if you go continuously
from one real number to another, you must pass through all the
real numbers in between. You can use it to prove the Ham Sandwich
Theorem; here’s how it can be used to solve Puzzles 1 and 2:
1. hiking the cascade Range. Solution. Puzzle 1 asked us to
prove that the programmers who spent
Saturday climbing and Sunday descending Mt. Baker were, at some time
of day, at exactly the same altitude on
It’s easily done. For any time t, let
f(t) be the progammers’ altitude on
Sunday minus their altitude on Saturday; f(t) starts off positive in the morning and ends up negative at night, so at
some point must be 0.
An equivalent, and perhaps more
intuitive, way to see this is to imagine
that the programmers have twins who
were instructed to climb the mountain
on Sunday exactly as the programmers
climbed it the day before. Then, even
if their paths up and down were different, there is some point at which the
programmers and their twins must
pass one another in altitude.
2.inscribing a Lake in a square.
Solution. Puzzle 2 asked us to show
that, given any closed curve in the
plane, there is a square containing the curve, all four sides of which
touch the curve. The idea of the proof
is both simple and elegant. Start
with a vertical line drawn somewhere
west of the curve. Gradually shift the
line eastward until it just touches
the curve. Repeat with a second line,
drawn east of the curve and moving
gradually west, so we now have another vertical line touching the curve
on its east side. Now bring a horizontal line down from the north until it
touches the curve and another from
the south, thus inscribing the curve in
But what we want is not merely a
rectangle but a square. Suppose the
rectangle is taller than it is wide (as it
would be in, say, Lake Champlain).
Now slowly rotate the four lines together clockwise, keeping all four outside
but still touching the curve. After 90 degrees of rotation, the picture is exactly
the same as before, only now, the previously long vertical lines of the rectangle are the short horizontal sides.
At some point in the rotation process, the original vertical lines and horizontal lines were all the same length—
and, at exactly that point, the curve was
inscribed in a square.
3.curves containing the corners of a square.
Solution. The third puzzle was (as
usual) unsolved, frustrating geometers for more than a century. For a
discussion see http://www.ics.uci.
html, including reference to an article
by mathematician Walter Stromquist
(“Inscribed Squares and Square-like
Quadrilaterals in Closed Curves,”
Mathematika 36, 2 (1989), 187–197)
in which he proved the conjecture for
smooth curves. See also Stan Wagon’s
and Victor Klee’s book Old and New
Unsolved Problems in Plane Geometry
and Number Theory (Mathematical Association of America, 1991).
All readers are encouraged to submit prospective
puzzles for future columns to firstname.lastname@example.org.
Peter Winkler ( email@example.com) is Professor of
Mathematics and of Computer Science and Albert bradley
Third Century Professor in the Sciences at Dartmouth
College, Hanover, nH.