Sunday Times Teaser 2939 – Betting to Win
by Mike Fletcher
Published January 20 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?
One Comment
Leave one →
-
Brian Gladman permalink12345678910111213141516171819202122232425262728# Let the bets on an A win be A, B on a B win and D on a# draw. Irrespective of the outcome £1 is won. Hence:## if A wins: (8/5).A - B - D = 1# if B wins: (6/5).B - A - D = 1# if a draw: (X/Y).D - A - B = 1## These equations can be solved giving (for an integer k):## A = 55.k, B = 65.k, D = 23.k - 1## (23.k - 1).X = (120.k + 1).Y## k = (X + Y) / (23.X - 120.Y)## For k > 0, Y < (23/120).X ==> Y_max < 4 if X_max < 20k = 0while True:k += 1for Y in (1, 2, 3):X, r = divmod((120 * k + 1) * Y, 23 * k - 1)if not r:print(f"£{23 * k - 1} was bet on a draw at odds of {X}-{Y}"f" (bets on wins: A £{55 * k}, B £{65 * k}).")exit()