»  Solutions to puzzles in my National Review Online Diary

  June 2005

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In my June diary I posed the following brain-teaser.

A person is bowling and has scores of 140, 85, 125, and 150. In his fifth round his score is within 3 of the average score for the first 5 rounds. How many possible medians are there for the set of 5 scores?

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Solution

Average of the five rounds is (140 + 85 + 125 + 150 + X) / 5, where X is the score for the 5th round.

This is equal to 100 + X / 5.  So I know that 97 + X / 5 ≤ X ≤103 + X / 5.

Subtracting X / 5 all through:  97 ≤ 4X / 5 ≤ 103.

Multiplying by 5 / 4 all through:  121¼ ≤ X ≤ 128¾.

Since a bowling score must be a whole number, the only possibilities are X = 122, 123, 124, 125, 126, 127, or 128.

Laying out the five-round scores in these seven cases, in ascending score order for each, I have:

85, 122, 125, 140, 150
85, 123, 125, 140, 150
85, 124, 125, 140, 150
85, 125, 125, 140, 150
85, 125, 126, 140, 150
85, 125, 127, 140, 150
85, 125, 128, 140, 150

The medians in the seven cases are 125, 125, 125, 125, 126, 127, 128. So there are four possible medians.

Answer:  4