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The number ³¹⁄₈₈ in the context of tennis

What has 3188 to do with tennis?

– an observation that is nearly trivial, but not quite

In a best‐of‑5‑set tennis match that provides for tiebreakers in all sets, you can win the match while winning only 3188 of the points. You cannot win a tennis match played under any standard set of rules while winning a smaller proportion of the points.

Specifically, you can win the match 0–6, 0–6, 7–6, 7–6, 7–6. The first four sets can of course take place in any order. The games you win, you win to 30. The games you lose, you lose to 0. The tiebreakers, you win 7–5. You win 93 points out of 264, a fraction of 3188. I omit the arithmetic.

That would be utterly boring were it not for a curious twist in the arithmetic. In what follows, the word minimal means “done in such a way as to win as small a proportion of the points as possible”. The words you and your refer to the winner of the match.

I asserted that the 3 sets that you win, you win 7–6 rather than 6–4. We cannot reach that conclusion on any kind of principle: we have to do the arithmetic. The interesting twist is that if there were such a thing as a best‐of‑7‑set match, then your minimal win would be 0–6, 0–6, 0–6, 6–4, 6–4, 6–4, 6–4, with scores of 6–4 rather than 7–6 in the sets you win. The reason is as follows. In a best‐of‑5‑set match with no tiebreakers, your minimal win is by 0–6, 0–6, 6–4, 6–4, 6–4, and you win 72 points out of 204, a fraction of about 0.353. In a best‐of‑7‑set match with no tiebreakers, your minimal win is by 0–6, 0–6, 0–6, 6–4, 6–4, 6–4, 6–4, and you win 96 points out of 280, a fraction of about 0.343. In either case, 5‑set or 7‑set, if you replace a set minimally won 6–4 by a set minimally won 7–6 then you introduce 20 more points, of which you win 7. That changes the proportion of points you win, by pulling it slightly towards 720 or 0.350. That is, downwards from 0.353 in the 5‑set case but upwards from 0.343 in the 7‑set case. The numbers are on a knife edge.