## Teaser 2916: Pointless batting averages

**From The Sunday Times, 12th August 2018** [link]

When my son was chosen to play cricket for his School First XI, I kept a record of the number of runs he scored in each of his first five innings. After each innings I calculated his batting average (the total number of runs scored so far divided by the number of innings) and found an interesting pattern:

(i) Each score was under 30

(ii) They [the scores] were all different

(iii) Each of the five averages was a whole numberWhen he asked me how he could maintain this pattern with his sixth innings, I was able to tell him the smallest score that would achieve this.

What is the largest number this could have been?

[teaser2916]

## Jim Randell 1:37 pm

on5 May 2019 Permalink |If the scores in the first five innings are

(a, b, c, d, e)and there are scores for the sixth inningsf, (between 0 and 29), that continue the pattern. And there will be a smallest such value:So, we can look at all possible

(a, b, c, d, e, f)values and find thelargestpossiblef_minvalue.This Python program runs in 569ms.

Run:[ @repl.it ]Solution:The largest number it could have been is 23.I found 142 sequences that give this largest possible

f_minvalue, although only 15 of these also have the averages take on 5 different values.One possible sequence is:

which give the corresponding averages of:

A score in the sixth innings of 23, would give an average of 18 over the six innings.

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