## Brain-Teaser 504

**From The Sunday Times, 7th February 1971** [link]

The inter-house Scrumps competition has just ended at Marlinghurst School. The four houses play each other once and the scoring is on a league basis (win 2, draw 1, lose 0). Arne played Bach in the 1st round, Chopin in the 2nd and Debussy in the 3rd.

In the 1st round no two houses scored the same number of “scrumps” (a scrump is a sort of goal scored by forcing an opponent through a hole in the wall).

In the 2nd and 3rd rounds every house scored exactly twice as many scrumps as its

currentopponent had scored in thepreviousround.Chopin and Debussy tied on points, but a better scrump average (i.e. ratio of scrumps

forto scrumpsagainst) gave Debussy the Pergolesi Bowl. Arne and Bach also tried, but Arne’s lower scrump average put them fourth in the final table.Although the total number of scrumps scored in the competition fell short of the school record (55) it was higher than last year’s total (44).

What was:

(a) Arne’s scrump average?

(b) the score in the Bach-Chopin match?

[teaser504]

## Jim Randell 9:00 am

on6 October 2019 Permalink |If we know the scores in the first round (say:

a, b, c, dscrumps, scored byA, B, C, Drespectively), then the scores in each match are:So the total number of scrumps scored is:

7(a + b + c + d).And this lies between 44 and 55, so it must be 49, hence

a + b + c + d = 7, and the individual totals must be 0, 1, 2, 4 in some order.This Python program runs in 86ms.

Run:[ @repl.it ]Solution:(a) Arne’s scrump average was: 12/17 (≈ 0.71); (b) The score in the Bach-Chopin match was 16 – 0.The scores in the three rounds were:

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