The seemingly simple act of dividing resources – whether it’s a poorly brewed pot of coffee or selecting teams for a competitive game – often leads to unfair outcomes. A surprising solution lies in a mathematical sequence first studied in the 19th century: the Thue-Morse sequence. This pattern, originally explored by mathematicians like Eugène Prouhet, Axel Thue, and Marston Morse, provides a method for fairer allocation when a straightforward “take turns” approach doesn’t suffice.
The Problem of Sequential Selection
Consider a pot of coffee brewed unevenly, with stronger concentration at the bottom. Pouring into two cups sequentially results in the first cup being weaker than the second. This illustrates a broader issue: when selecting from a ranked set of items (players, resources, etc.), the first selector gains an inherent advantage.
In a team-picking scenario, if one captain chooses all players first, leaving the remainder to the other, a severe imbalance in team quality emerges. Even alternating picks doesn’t fully correct this: the first selector consistently secures the stronger players, leading to a skewed distribution of talent. In a ranked set of players from 1 to 10, the first team will total 30, while the second only 25.
The Thue-Morse Solution: Taking Turns at Taking Turns
The Thue-Morse sequence offers a solution by introducing a nested turn-taking pattern. Instead of simply alternating (AB, AB), it employs a sequence of ABBA. The first pair follows the standard order, but the second reverses it. Repeating this pattern (ABBA BAAB BAAB ABBA) ensures a more balanced distribution.
Applying this to the team-picking example, the final scores become 27 and 28, a significant improvement over the 30 vs 25 imbalance. This approach levels the playing field by preventing one selector from consistently acquiring the highest-ranked items.
Real-World Applications and Fairness
The Thue-Morse sequence isn’t just a mathematical curiosity; it’s used in real-world scenarios where fairness is paramount. Tennis tie-breaks employ a simplified version (ABBA) where players alternate serving two consecutive points, ensuring equal pressure. FIFA and UEFA have even trialed it for penalty shoot-outs, adding pressure to the second shooter in each pair.
For the uneven coffee pot, the solution is elegant: pour half a cup into the first, two halves into the second, then the final half back into the first. The result: two cups of equal strength. While stirring achieves the same outcome, the mathematical approach adds a satisfying layer of precision.
The Thue-Morse sequence demonstrates that fairness isn’t always intuitive. Sometimes, a little 19th-century math is all it takes to level the playing field and ensure a more equitable outcome.
