In Search of Woodall Primes

Episode Summary

Executive Summary: The episode pairs a playful case for celebrating overlooked Woodall primes with a maths-of-sport segment on making penalty shootouts fairer. Matt Parker argues that prime-number subfamilies deserve more attention and enlists distributed computing to find new Woodall primes, while the second half explains why shootouts favor the first kicker and why ABBA/Toeplitz-Morse-style sequences aim to reduce that bias, though practicality still limits adoption.

Main Topics: Matt Parker’s campaign for Woodall primes (Priority: 5/5): Parker urges listeners to pay attention to Woodall primes, a lesser-known prime family similar to Mersenne primes but defined by n×2^n−1, and frames 2017 as a centenary opportunity to celebrate them. Mersenne primes and distributed computing (Priority: 4/5): The conversation contrasts Woodall primes with the more famous Mersenne primes, noting how GIMPS and computer searches dominate discoveries of very large primes and serve as a testbed for software reliability. Definitions and mathematical structure of primes (Priority: 4/5): Harford and Parker clarify what primes are, why special prime families matter, and how patterns around powers of two generate numbers with striking factor properties before a prime appears nearby. BOINC/PrimeGrid as a search tool for Woodall primes (Priority: 4/5): Parker recommends PrimeGrid within BOINC as a way for volunteers to donate unused computing power to search for neglected prime types, even though BOINC also supports more socially useful projects. Penalty shootouts and first-kicker advantage (Priority: 5/5): The World Service segment explains that standard shootouts are considered unfair because the team kicking first appears to have about a 60% chance of winning, motivating changes to the sequence. ABBA and the Tue-Morse sequence (Priority: 4/5): A revised shootout order (ABBA) and a more complex Tue-Morse sequence are presented as mathematical attempts to balance psychological pressure and fairness, with an analogy to fair coffee sharing. Limits of mathematical fixes in real sport (Priority: 3/5): Despite elegant mathematical proposals, the segment concludes that complexity and logistics make these systems hard to implement fully, and real-world results may still not remove the first-mover advantage.

Key Arguments: Woodall primes deserve attention because mathematics often advances by noticing overlooked patterns, not just famous examples like Mersenne primes. Mersenne primes are easier to search for computationally, which is why the largest known primes are usually of that form. Prime search projects are also valuable for debugging computers because large-prime verification requires cross-checking software and hardware. Woodall primes generalize a simple pattern: n×2^n−1 sometimes produces primes, and mathematicians study why this happens and how it extends to other bases. Penalty shootouts are statistically biased toward the team that goes first, with cited research estimating a roughly 60% win rate for first kickers. ABBA reduces some of the bias by alternating who gets the first kick in paired penalties, but it may still leave subtle advantage in odd/even rounds. The Tue-Morse sequence is mathematically elegant and fairer in theory, but its complexity makes it unlikely to be adopted in professional football. Mathematical sequences can improve fairness in other settings too, such as dividing coffee with uneven concentration. A practical compromise often wins over a theoretically superior but cumbersome method when human memory and usability matter.

Data Points: Mersenne prime digits: 22 million digits - Referenced as the newly discovered record prime nicknamed Mersenne 49 Largest known prime discovery location: Missouri - Mentioned as the place where the new record prime was found Prime discovery system: GIMPS (Great Internet Mersenne Prime Search) - Distributed computing project that found the record Mersenne prime Woodall prime example: 383 - Parker’s favorite Woodall prime and a palindromic base-10 number Smallest Woodall prime: 7 - Derived from 2^2×2−1 Next Woodall prime after 7: 23 - Derived from 2^3×3−1 Largest Woodall prime mentioned: 1,129,757 digits - The biggest Woodall prime found, discovered 10 years earlier Woodall centenary: 1917 - Woodall and Cunningham’s paper marking 100 years of study at the time of the segment Shootout win probability for first team: around 60% - Statistics cited from 2010 on penalty shootout advantage Standard shootout pattern: 5 penalties each, then sudden death - Described as the current system before proposed changes Liverpool vs Middlesbrough shootout: 30 penalties - 2014 domestic cup match ended 14-13 England vs Holland U21 shootout: 32 penalties - 2007 semi-final ended 13-12 ABBA sequence: A, B, B, A - Proposed revised penalty order to balance first-kick advantage Tue-Morse example sequence: ABBA, BAAB, BAAB, ABBA - Illustrated as a more mathematically balanced sequence Coffee-sharing example: 8 divisions - Used to show Tue-Morse can fairly distribute uneven coffee concentration Women's U17 Euro final: 14 May - ABBA system was tested in a real shootout; Germany went first and won

Pivotal Quotes: "I think there are some unsung primes that aren't getting the attention they deserve." — Matt Parker: Parker explains why he wants listeners to care about Woodall primes "Why not just keep switching around the advantage of who goes first?" — Ignacio Palacio Suerta (via narrator): Simple rationale behind the ABBA penalty-kick proposal "The thing is, I don't know anything useful, but it's just this mathematicians wanting just to explore and look at these patterns and just discover things partly for the sake of it." — Matt Parker: Why pure mathematical curiosity still matters

Implications: Listeners get a reminder that obscure mathematical patterns can be both beautiful and practically useful, from computing to sports fairness. But the football segment also shows that elegant math often yields to human factors and operational simplicity.