Did you know that you could lose a single transferable vote election if you convince supporters of an opponent to vote for you instead?

Suppose Tom, Dick and Harry are contesting such an election and one seat is up for grabs. To keep numbers small for this example, suppose there are 27 voters. Therefore the winner will be the first who gets 14 votes, either as first count votes or as first count plus inherited votes after a candidate is eliminated.

Suppose the voting for this example is as follows. Ten voters express their preferences as 1. Tom, 2. Dick, 3. Harry, nine voters express their preferences as 1. Dick, 2. Harry, 3. Tom, and eight voters vote 1. Harry, 2. Tom, 3. Dick. The first count votes are therefore 10 for Tom, nine for Dick and eight for Harry, who is therefore eliminated. Harry’s eight votes are inherited by Tom, who gets 18 votes and is therefore elected.

But suppose that in the week before election day Tom convinces two of Dick’s supporters to vote for him so that now, the voting preferences look like this: 12 voted 1. Tom, 2. Dick, 3. Harry, seven voted 1. Dick, 2. Harry, 3. Tom and eight voted 1. Harry, 2. Tom, 3. Dick. Therefore now Dick will be eliminated after the first count, Harry will inherit his seven votes and, with a total of 15 votes, gets elected instead of Tom.

Can you count?

In a party of people some know each other and others do not. We assume that if A knows B then B knows A. Every person knows exactly three other people. No two acquaintances have a common acquaintance, while any two persons who do not know each other have exactly one common acquaintance. How many persons are there at the party?

Solution: 10.