Puzzled: circular food
Welcome to three new challenging mathematical puzzles. Solutions to the first two will be published
next month; the third is as yet unsolved, so you may need extra luck with that one. Here, I concentrate
on circular food, so you might want to eat something before jumping in.
1.Tw o hungry Icelanders, Alta and Baldur, are
sharing a pizza that is radially sliced into pieces
of various sizes. To start, Alta chooses any slice to
eat; thereafter, she and Baldur alternate taking slices
(Baldur first) but must always take a slice that is
adjacent to some previously taken slice. I call this the
“polite pizza protocol”; the uneaten portion is always
connected, thus, in theory, staying hot longer. Is it
possible for the pizza to be cut in such a way that, no
matter what Alta does, Baldur can get more than half
2.A cylindrical ice-cream cake with the most
scrumptious chocolate frosting on top is sitting on
a table. As an expert cake cutter, you choose an arbitrary
angle x and proceed to cut one wedge after another,
counterclockwise, around the cake, each of angle exactly
x. However, each time you cut a wedge, you turn that
piece upside-down and slide it back into the cake. This
puts the frosting on the bottom at first, but as you work
your way around and around the cake, the frosting
comes back up to the top, then returns to the bottom,
and so forth. Your mission is to prove that after some
finite number of slices, all the frosting will be back on
top of the cake.
3.You need to bake circular tarts of various sizes and
total area 1. Can you always fit them into a circular
pie pan of area 2? Amazingly, no one knows.
(this puzzle is from alexander soifer of the university of Colorado, Colorado springs; related work, including a proof for
square tarts in a square pan, can be found in the journal Geombinatorics, www.uccs.edu/~geombina/.)
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.