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Quantum Physics Finds a Loophole in Euler's 6x6 Officer Problem

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Quantum Physics Finds a Loophole in Euler’s 6x6 Officer Problem

Clip title: Quantum Physics Found a Loophole in an IMPOSSIBLE Math Problem Author / channel: Up and Atom URL: https://www.youtube.com/watch?v=SCI8g6EinAo

Summary

The video introduces a classic mathematical puzzle: arranging 36 unique symbols (each with one of six colors and one of six shapes) into a 6x6 grid such that no color or shape repeats in any row or column. Classically, this problem, known as Euler’s Officer Problem, is impossible to solve. However, the video highlights a recent paper proposing a “quantum solution” to this classically impossible problem, delving into the fascinating intersection of mathematics and quantum physics.

The historical roots of Euler’s Officer Problem trace back to 1779 when Catherine the Great challenged mathematician Leonhard Euler to arrange 36 military officers (each defined by a unique rank and regiment combination) in a 6x6 formation following these non-repetition rules. Euler attempted to solve it and concluded it was impossible. The problem is a specific instance of a “Graeco-Latin Square” where two distinct attributes (like color and shape) must be uniquely placed within rows and columns. While solutions exist for other sizes (like 3x3, 4x4, 5x5, and even larger squares like 22x22, which disproved Euler’s initial general conjecture that all 4k+2 size squares were impossible), the 6x6 case remained stubbornly unsolvable in classical mathematics, a fact later confirmed by Gaston Tarry in 1901 after exhaustively checking all possibilities.

Intriguingly, the number “6” also presented a unique challenge in the realm of quantum physics, specifically concerning “Absolutely Maximally Entangled” (AME) states. An AME state describes a system of quantum particles where every possible pair of particles is maximally entangled, meaning the state of any two particles can perfectly infer the state of any other two. These states are crucial for quantum computing and information theory, particularly for error correction. For systems of four quantum particles, AME states are known to exist for most numbers of possible “states” each particle can have, except for 2 and 6. The striking parallel between the problematic sizes (2 and 6) in both classical Graeco-Latin squares and quantum AME states suggests a deeper mathematical connection.

The quantum solution involves “quantifying” the classical problem by introducing new rules based on quantum principles like superposition and entanglement. This process transforms the discrete attributes of classical officers into a continuous spectrum of possibilities, offering more “freedom” in arranging them. The researchers redefined the rules: each quantum officer must be “perfectly distinguishable” (Rule 0), and all colors (Rule 1) and shapes (Rule 2) must be “equally balanced” across every row and column in terms of quantum probabilities. Using a computer algorithm to navigate this vast, infinite space of quantum possibilities, they found a valid 6x6 Quantum Graeco-Latin Square. A surprising byproduct of this solution was the appearance of the Golden Ratio in the quantum probabilities, leading them to coin the term “Golden AME State.” This remarkable discovery highlights how mathematical problems, once thought impossible in one domain, can find elegant solutions by shifting to a more expansive, quantum framework, demonstrating the immense progress in scientific understanding over centuries.

Video Description & Links

Description

Watch my exclusive documentary Is Math Invented or Discovered? https://nebula.tv/ismathinvented

Up and Atom on Spotify https://open.spotify.com/show/033gE0X74RZAwIh3Tbj6B7

A big thank you to my AMAZING PATRONS! Michael Seydel, Brian Wilkins, Thorsten Auth, Chris Flynn, Richard O McEwen Jr, Scott Ready, Izzy Ca, John H. Austin, Jr., David Johnston, Thomas Krause, Ave Eva Thornton, Anne Tan, abc, Nahuel Hernán Espiño, Norman Jaffe, Jim Wright, Elliot Glazer, Cumaco, John Doe, Dirk Wouters, Stoney Bair, Francisco, Marc-Antoine, Thomas Urech, chuck zegar, David Tuman, Ben Mitchell, Jim Felich, Jeremy, DONALD McLeod, Ron Hochsprung, James Matheson, Tim Ludwig, Dagmawi Elehu, Jeffrey Smith, Alex Hackman, Joel Becane, Paul Barclay, John Lakeman, Jana Christine Saout, Jeff Schwarz, Yana Chernobilsky, Michael Dean, Chris Amaris, Matt G, Dag-Erling Smørgrav, John Shioli, Gladamas, Hey kid I’m a computer, Bli Blop, Steve Archer, Antonio Gonzalez, Robert Ball, AdamM, Dhandforth, Wolf Witt, Benedict, Janusz Górecki, Prashant Savalia, Henrik Horneber, Markus S., AudioFreak, Lauren Fish, Matthew Lewis, Pawel W, Bryan Hughes, Trance Maharaj, Matt, Jonathan Rayback, Smarter Every Day, Bunny Lushington, Jane!, Motty Porat, Carlos Escolar, Keenir, robert lalonde, Cassandra Durnord, Paul Bunbury, Richard Rensman, David Shlapak, Kent Arimura, Phillip Rhodes, Michael Nugent, Roland Gibson, Joe McTee, Oleg Dats, Simon J. Dodd, Tang Chun, William Toffey, omg.science, Cameron Tacklind, Lance Ahmu, Steve Watson, Andi B, Thomas P Taft, Pablo de Caffe, Taylor Hornby, Colin Byrne, Spuddy, Sascha Bohemia, Stephen Britt, KG, Hansjuerg Widmer, John Sigwald, O C, Carlos Gonzalez, James Palermo, Thomas V Lohmeier, Chris Teubert, Wolfgang Ripken, Nicolas Frias, Louis M, ROBERT C PAYNE, Rick DeWitt, Gene Levitsky, Pat Gunn, RobF, Vincent Seguin, Michael Brunolli, Shawn, Steven Wheeler, Philip Freeman, Jareth Arnold, and Simon Dargaville.

Creator - Jade Tan-Holmes Script - Daniel Inafuku, Elizabeth Fernandez, and Jade Tan-Holmes Music - epidemicsound.com

9 × 4 = 6 × 6: Understanding the quantum solution to the Euler’s problem of 36 officers https://arxiv.org/abs/2204.06800

Chapters 0:00-1:05 Intro 1:05-1:59 Euler’s Officer Problem 1:59-2:25 Latin Squares 2:25-3:29 Graeco-Latin Squares 3:29-5:20 Euler’s Conjecture 5:20-6:54 Quantum Basics 6:54-8:59 AME States 8:59-10:24 woahhhh 10:24-11:40 Lifting 11:40-14:34 Quantum Euler’s Officer Problem 14:34 Golden AME State

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