Enigma 1422: Incentive
11 June 2013
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From New Scientist #2582, 16th December 2006
The four football teams, Magdala, Nazareth, Sepphoris and Tiberias, play each other once a season, with 2 points for a win and 1 for a draw. At the end of the season there is a table which orders the teams by points scored, and teams with equal points are bracketed together.
The prize money, in drachmas, is 40, 30, 20 and 10 for the teams in 1st, 2nd, 3rd, 4th places respectively; when teams are bracketed they divide the total money for their places evenly between themselves.
One match is played each week with the first four being MvS, NvT, MvT, NvS. After the fourth match, Jesus works out the points totals situation (PTS) and uses it to determine whether the fifth match should be MvN or SvT. He makes the choice so that in the sixth match both teams have a financial incentive to play hard. For example, if the PTS is M3, N2, S1, T2 then whichever is the fifth match, both teams in the sixth will play hard.
For how many PTSs is it essential to have MvN as the fifth match?