## Teaser 2939: Betting to win

**From The Sunday Times, 20th January 2019** [link]

Two teams, A and B, play each other.

A bookmaker was giving odds of 8-5 on a win for team A, 6-5 on a win for team B and X-Y for a draw (odds of X-Y mean that if £Y is bet and the bet is successful then £(X + Y) is returned to the punter). I don’t remember the values of X and Y, but I know that they were whole numbers less than 20.

Unusually, the bookmaker miscalculated! I found that I was able to make bets of whole numbers of pounds on all three results and guarantee a profit of precisely 1 pound.

What were the odds for a draw, and how much did I bet on that result?

[teaser2939]

## Jim Randell 8:17 am

on6 March 2019 Permalink |If the integer stakes are

pfor A to win,qfor B to win,rfor a draw, we have:which, given

XandY, we can solve as:and we want these to be integers.

XandYare limited to be integers between 1 and 19, so we can easily check all possibilities.This Python program runs in 72ms.

Run:[ @repl.it ]Solution:The odds for a draw were 11-2. The bet on a draw was £22.A variation: If the bets can be whole numbers of pennies, instead of whole numbers of pounds, then there is another solution where X-Y is 10-1 and the bets are £5.50 on A to win, £6.50 for B to win, £1.30 on a draw. The winnings are £14.30 for a total outlay of £13.30.

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## GeoffR 8:22 am

on7 March 2019 Permalink |LikeLike