# Solution to Enigma No. 1522

**Answer:** Twente beat Sporting; Sporting 9 points, Twente 5 points

The winner is Richard Kennaway of Norwich, UK. There were 77 entries.

## Worked answer

There must be at least 4 points between successive teams. Total points gained after 5 weeks are 30 if no matches have been drawn, 29 if 1 has been drawn, 28 if 2 have been drawn, etc.

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If R have 12 points, points scores must be R12, S8, T4, U0. This gives a total score of 12+8+4+0 = 24, requiring 6 drawn matches – impossible since neither R nor U can have drawn any. So R have 15 (5 w) or 13 (4w, 1d) points.

**If R have 15 points**, S have played R at least once and lost, so cannot have more than 10 points (3w, 1d, 11); in that case R and S have 25 points between them, and T can only have 4 points (1w, 1d, 31) and U 0 points (51): 15+10+4+0 = 29, with SvT the 1 drawn match.

If R have 15 points and S 9 points (3w, 21), T cannot have 5 points (1w, 2d, 21) because 15+9+5 = 29, which allows only 1 drawn match.

Nor can T have 4 points and U 0 points: 15+9+4+0 = 28, requiring 2 drawn matches; but with these points only T can have drawn a match.

Nor can the points scores be R15, S8, T4, U0: 15+8+4+0 = 27, requiring 3 drawn matches, but with these points neither R nor U have drawn any.

So if R have 15 points, the scores are **R15, S10, T4, U0**. R have won all 5 matches and S have lost only once, so RvS and TvU are outstanding. The results are:

(1) R beat S, R beat T (twice), R beat U (twice); S beat T, SvT drawn, S beat U (twice); T beat U.

**If R have 13 points**, points scores cannot be R13, S8/9, T4, U0 because 13+8/9+4+0 = 25/26, requiring 5/4 drawn matches – impossible since R have drawn only 1 and U haven’t drawn any. The number of drawn matches is manageable only if points scores are R13, S9, T5, U0/1.

With **R13, S9, T5, U0**, since 13+9+5+0 = 27, S must have drawn 3 matches; neither R nor S have lost, so RvS and TvU are outstanding.

(2) **RvS drawn**, R beat T (twice), R beat U (twice); SvT drawn (twice), S beat U (twice); T beat U.

With **R13, S9, T5, U1** there must have been 2 drawn matches: RvT and TvU; there are 3 possibilities for the results so far, dependent on which matches are still outstanding:

(3) RvS, TvU outstanding: R beat S, R beat T, RvT drawn, R beat U (twice); S beat T, S beat U (twice); **T beat S**, TvU drawn.

(4) RvT, SvU outstanding: R beat S (twice), RvT drawn, R beat U (twice); S beat T (twice), S beat U; T beat U, TvU drawn.

(5) RvU, SvT outstanding: R beat S (twice), R beat T, RvT drawn, R beat U; S beat T, S beat U (twice); T beat U, TvU drawn.

The only unique results in the 5 possible sets are highlighted in bold:

RvS drawn in (2), T beat S in (3). Since the puzzle says that I would say **who beat whom**, it must be that T beat S as in (3). So S currently have 9 points and T 5 points.