The 19th-century maths that can help you deal with horrible coffee

Imagine that you have a pot of coffee that equals two cups. It is not brewed well, so it is much stronger at the bottom than at the top. If you pour coffee from a saucepan into two cups, the first one you pour will be much weaker than the second.

While this is a bit of a contrived situation, there are other cases where this “worse first” (or “better first”) attitude creates injustice.

Let's say we're choosing teams for a football match, and everyone knows roughly which players are better than others. If you allow one team captain to pick all of his players first and let the other captain choose whoever is left behind, there will be a major imbalance in how good the teams are.

Even just taking turns choosing doesn't make it fair: if there were players whose skills could be rated roughly from 1 to 10, then Captain A, choosing first, would choose 10, then Captain B would choose 9, then Captain A would choose 8, and so on. In total, the team that picks first will have 10 + 8 + 6 + 4 + 2 for a total of 30, and the other team will have 9 + 7 + 5 + 3 + 1 for a total of 25.

So how can we distribute players fairly? The answer to this question is provided by a mathematical sequence from the 19th century. Originally studied by Eugène Proué in the 1850s, but then described in more detail by Axel Thue and Marston Morse in the early 20th century, the Thue-Morse sequence requires you to not just take turns: you take turns.

Let's say two collectors are called A and B. Then the sequence will be like this: ABBA. The first pair is in one order, and the second pair is in the opposite order. If we want to continue the sequence, we can repeat the same set again, but swapping A and B: ABBA BAAB. This can be continued (in turns, in turns, in turns), giving ABBA BAAB BAAB ABBA and so on.

This arrangement makes things fairer. In our team matching example, instead of 30 versus 25, the teams are now 10 + 7 + 5 + 4 + 1 and 9 + 8 + 6 + 3 + 2, for a total of 27 and 28.

Variations of this sequence are often used in real sports competitions. Tiebreakers in tennis involve one player scoring the first point, then players take turns giving away two points in a row, in an ABBA, ABBA, ABBA pattern. This simplified version of Thue-Morse is considered fairer than just taking turns. A similar arrangement has been tried by FIFA and UEFA for penalty shootouts in football, where the second shot of each pair puts more pressure on the kicker.

For our coffee pot, the solution is ideal: pouring half a cup of coffee into cup A, then two half cups into B, and then the last half cup back into A, we get two cups of completely equal strength. If you prefer, you can simply stir the coffee with a spoon. But wouldn't it taste better if you used mathematics solve the problem?

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Kathy Steckles is a mathematician, teacher, YouTuber and author from Manchester, UK. She is also a consultant for New scientistBrainTwister puzzle column. Follow her @stecks