Week 4
October 12, 2015
1. Can two dice be weighted such that the probabilities of rolling a 2, a
3, . . . , and a 12 are all equal?
2. If I cram infinitely many people into a room and randomly put red
and blue hats on their heads (I have lots of both), and then get them
to all simultaneously guess what colour hat they are wearing, is there
a strategy such that infinitely many of them will be right? Assume
they have infinite vision and can see all hats except their own. Also
assume that an infinite number of people speaking simultaneously has
sufficiently finite volume that I don’t go deaf.
3. If every pair of people at a party have exactly one friend in common,
show that there is one person who is a friend to all things. Or at least,
to everyone at the party.
4. Find all functions f : Z → Z such that f (x + y) = f (x) + f (y). Do we
get any new solutions if we replace Z by Q? What about R?
5. Find a finite set S of points in the plane such that, given any p ∈ S,
there are exactly 3 points in S at distance 1 from p.
6. For any m ∈ N, prove there exists a finite set S of points in the plane
such that, given any p ∈ S, there are exactly m points in S at distance
1 from p.
7. Find the angle ∠CAD
8. What’s wrong with this illustration of Pythagoras
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