When you go to an MTC Workshop, occassionally the facilitator will put up what we call a "side board" problem. These are not the focus of a session, but just interesting problems for participants to think about and discuss. Sometimes participants solve them during the session and sometimes not. We never post the solutions though!
Planet Carefully has
only one landmass, a small island called Isle of Puzzles. On this
island is a fearless pilot who wants to fly all the way around his
watery planet. Unfortunately, planes on Planet Carefully can only
carry enough fuel to get halfway around the globe. Fortunately,
pilots can instantaneously transfer any amount of fuel from the tank
of one plane to that of another mid-flight. What is the smallest
number of planes that must participate in this adventure so that one
pilot flies without stopping around the globe and all pilots make it
In each round of an
elimination tournament, players are randomly paired and each pair
plays one game. Losers are eliminated from the tournament, but
winners go on to the next round. If there are an odd number of
players during a round, then one player does not play that round,
but does advance to the next round. So, for example, if there are 19
players, there will be 9 games played and 10 people will go on to
the next round. This continues until there is a single winner. If
there are n players to begin
with, how many games will
be played before a winner is determined? (Note that we are counting
games played as opposed to rounds played.)
Cossack brothers sold a herd of sheep. Each sheep sold for as many
rubles as the number of sheep originally in the herd. The money was
then divided in the following manner: first the older brother got
10 rubles, then the younger brother got 10 rubles, and so on. At
the end of the division, the younger brother, whose turn it was, saw
that there were fewer than 10 rubles left, so he took what remained.
To make the division fair, the older brother gave the younger one
his knife which was worth an integer number of rubles. How much was
the knife worth?
Problem 1: Pick any integer greater than or equal to 1. Multiply it by 9. Permute its digits in any way. Show that the number you get is a multiple of 9.
Problem 2: "The Infected Checkerboard" An infection spreads among the squares of an n by n checkerboard in the following manner: If a square has 2 or more infected neighbors, then it becomes infected itself. Neighbors are orthogonal only, so each square has at most 4 neighbors. Show that you cannot infect the whole board if you begin with fewer than n infected squares.