19th Century Math Tips for Taming Bad Coffee

Picture having a coffee pot that serves two cups. Poor brewing might result in a stronger brew at the bottom than at the top. When pouring from the pot into two cups, the first cup will taste much weaker than the second.

While this scenario is somewhat contrived, there are other situations where a “first is worse” (or “first is better”) approach can lead to inequity.

Consider a football game where everyone has a good idea of the skills of each player. If one team’s captain selects all players first, it creates a significant imbalance in team strength.

This scenario remains unfair even with a simple pick order. For instance, if players can be ranked from 1 to 10 based on skill, if Captain A chooses player 10 first, then Captain B selects player 9, followed by Captain A picking player 8, and so on, the resultant totals are skewed. Captain A’s team ends up with a score of 30 (10 + 8 + 6 + 4 + 2), while Captain B’s team scores only 25 (9 + 7 + 5 + 3 + 1).

So, how can we ensure a fair player selection? The answer lies in a mathematical method from the 19th century. The Tew-Morse series, initially explored by Eugène Plouet in the 1850s and subsequently detailed by Axel Tew and Marston Morse in the early 20th century, advocates for alternating and rotating choices.

In a scenario with selectors A and B, the selection order follows an ABBA pattern. The first pair is in the same order, while the second flips the order. This pattern can be extended, with a repeat that reverses the As and Bs: ABBA BAAB. Further sequences can be created like “ABBA BAAB BAAB ABBA”.

This rotation helps create equity. Using the team selection example again, the totals would be much more balanced: 10 + 7 + 5 + 4 + 1 for one team versus 9 + 8 + 6 + 3 + 2 for the other, leading to totals of 27 and 28.

An iteration of this sequence is also employed in sporting events. For instance, during a tennis tiebreak, one player serves first, followed by each player taking turns to serve two points in an ABBA sequence. This streamlined version of Tew-Morse is often seen as fairer than simple turn-taking. A similar approach is being tested by FIFA and UEFA during soccer penalty shootouts, applying pressure on the second shooter in each pair.

Returning to the coffee pot scenario, the solution is ideal. If you pour half a cup into cup A, then pour two half cups into cup B, and finally add the last half cup back into A, you will achieve equal strength in both cups. Alternatively, you could stir the coffee with a spoon. However, wouldn’t it be more gratifying to tackle such challenges with the aid of mathematics?

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katie steckles – A mathematician, lecturer, YouTuber, and author based in Manchester, UK. She also contributes to New Scientist‘s puzzle column “BrainTwister”. Follow her @stex