one to one
a correspondence column

Dear Sir,

The page on non-transitive dice (see MANIFOLD-10) prompts me to send you the following non-transitive betting game which is (to me) even more counter-intuitive. So far as I know It is not published. It was discovered by a man with the appropiate name of Walter Penney. and has been circulating by word of mouth. You ask your opponent to choose any triplet of heads and tails. You then announce your own choice of a triplet. A penny is flipped until one of the triplets occurs, and the person who chooses that triplet wins. Whatever triplet he chooses, you can choose a better one, according to the table below, which gives you the best odds.

   His choice    Your choice    Odds in your favour 
       HHH               THH                   7 TO 1 
       HHT                THH                   3
       HTH                 HHT                  2 
       HTT                 HHT                  2 
       THH                TTH                   2 
       THT                 TTH                   2 
       TTH                 HTT                   3 
       TTT                  HTT                   7 

The table was worked out for me by N.S. Mendelsohn.

Martin Gardner

Dear Sir,

I do not usually bother to send you comments on the various combinatorial articles that appear in MANIFOLD, but I thought I would just remind you of the following theorem, which you appear to have temporarily forgotten.

THREE COLOUR THEOREM
A trivalent map in the plane can be properly coloured with three colours if and only if every region has an even number of sides.

To prove this result one constructs the dual map and then applies:

TWO COLOUR THEOREM
A map in the plane can be properly coloured with two colours if and only if every vertex has an even number of edges.

This is not intended as an entry for the Great MANIFOLD Contest (M-12), as it is hardly in the right spirit. Perhaps it will get a cordial reception.

Douglas Woodall