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

Imagine you have a coffee maker that is equivalent to two cups. It was brewed incorrectly, so it is much stronger at the bottom than at the top. If the coffee is poured into two cups, the first one you pour will be significantly weaker than the second.

While this is a slightly artificial situation, there are other occasions where this type of “first is worst” (or “first is best”) setup creates injustice.

Suppose we’re picking teams for a soccer game and everyone pretty much knows which players are better than others. If you allowed one team captain to pick all of his players first, leaving the other captain with the remaining one, there would be a serious imbalance in the quality of the teams.

Even just taking turns choosing doesn’t make it fair: if there were players whose skills could be roughly ranked 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. Overall, the team that picks first would have 10+8+6+4+2, making a total of 30, while the other would have 9+7+5+3+1, making a total of 25.

So how can we distribute players fairly? A 19th century mathematical sequence holds the answer. Originally studied by Eugène Prouhet in the 1850s, but later written about in more detail by Axel Thue and Marston Morse in the early 20th century, the Thue-Morse sequence requires that you don’t just take turns: you take turns.

Let’s say the two team selectors are named A and B. The sequence would then be: ABBA. The first pair is in the same order, but the second pair is in the reverse order. If we want to continue the sequence, we can repeat the same set again, but reversing As and Bs: ABBA BAAB. This can be continued (taking turns, taking turns), giving ABBA BAAB BAAB ABBA, and so on.

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

Versions of this sequence are often used in real sporting competitions. Tie-breaks in tennis involve one player serving the first point, then players take turns serving two consecutive points – giving the pattern ABBA ABBA ABBA. This simplified version of Thue-Morse is widely considered fairer than simple turn-taking. A similar order has been tested by FIFA and UEFA for penalty shootouts in football, where the second shot in each pair places greater pressure on the shooter.

For our coffee maker, the solution is perfect: pouring half a cup of coffee into cup A, then two half cups into B, then the last half cup into A, will give two cups of exactly equal strength. If you prefer, you can simply use a spoon to stir the coffee. But won’t it taste more satisfying if you used math to solve the problem?

These articles are published weekly on
newscientist.com/maker

Katie Steckles is a mathematician, speaker, YouTuber and author based in Manchester, UK. She is also an advisor for New scientistthe BrainTwister puzzle section. Follow her @stecks