Sunday Times Teaser 2998 – Football League
by Andrew Skidmore
Published Sunday March 08 2020 (link)
In our league three points are awarded for a win and one for a draw. Teams play each other twice per season. Comparison of the results of the two Divisions at the end of last season showed that
a. Division II held one more team than Division I.
b. The same total number of points was awarded in each division.
c. For the division with the larger number of draws (drawn matches), that number was equal to the number of matches not drawn in the other division.
d. The number of draws in one division was a multiple of the (3-digit) number of draws in the other.
How many teams were in Division II?
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Brian Gladman permalink123456789101112131415161718192021222324252627282930313233# Division 1: n teams, n.(n - 1) matches: u won/lost, d drawn# Division 2: n + 1 teams, n.(n + 1) matches: v won/lost, e drawn## u + d == n.(n - 1) (1)# v + e == n.(n + 1) (2)## Points: 3.u + 2.d == 3.v + 2.e, then using (1) and (2)## u == v + 4.n (3)# e == d + 6.n (4)# u + e == n.(n + 5) (5)## Clause (c) and (d): e == u ; e = m.d## u = e = n.(n + 5) / 2 (6)## m == (n + 5) / (n - 7) (7)# n == 7 + 12 / (m - 1) (8)## Hence m - 1 = 1, 2, 3, 4, 6, 12# the clause C multiplierfor i in (1, 2, 3, 4, 6, 12):n = 7 + 12 // iu = e = n * (n + 5) // 2d = e - 6 * nv = u - 4 * n# the lesser number of drawn matches is a three digit numberif 100 <= d < 1000:for n, t, p, w, d in ((1, n, n * n - n, u, d), (2, n + 1, n * n + n, v, e)):print(f"Division {n}: {t} teams in {p} matches: {w} win/loss, {d} drawn.")
