Donald Knuth's 2025 Christmas Lecture: Exploring the Knight's Tours

In early December 2025, Donald Knuth, the esteemed professor emeritus at Stanford University, delivered his annual Christmas lecture, a tradition he has upheld for over 30 years. As he approaches his 88th birthday, Knuth continues to work on his seminal work, "The Art of Computer Programming," while this year's lecture focused on a long-standing problem in graph theory: the Knight's Tour on a chessboard, which asks if a knight can visit every square without revisiting any.

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Knuth's fascination with the Knight's Tour began in 1973, and he has since revisited the problem numerous times. During the lecture, he shared insights from his recent research, revealing that he has calculated the total number of possible solutions to the Knight's Tour problem to be 13,267,364,410,532. This figure was first computed by Australian mathematician Brendan McKay in 1997, but Knuth's work has provided new classifications and insights into the nature of these tours.

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Knuth's exploration of the Knight's Tour includes identifying the angles formed during the knight's moves. He noted that the maximum number of right angles in a solution has increased from 38 to 39, and he detailed the various configurations that can yield different angles, including acute and obtuse angles. His findings indicate that there are 28,000 solutions that avoid acute angles entirely, while only one solution exists that contains exactly four obtuse angles.

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Throughout the lecture, Knuth emphasized the aesthetic appeal of mathematical solutions. He described the Knight's Tours as akin to art, where the beauty of the patterns formed by the knight's movements can be appreciated visually. He shared examples of "whirling" Knight's Tours, which exhibit symmetry and intricate designs, culminating in a stunning 18 x 18 tour that maintains its form when rotated 90 degrees.

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The lecture concluded with a Q&A session, where Knuth engaged with the audience, sharing anecdotes and insights into his lifelong passion for mathematics. His Christmas lectures have become a beloved tradition at Stanford, capturing the interest of both students and seasoned mathematicians alike. Recordings of past lectures have been restored, allowing a broader audience to appreciate Knuth's contributions to computer science and mathematics.

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Knuth expressed his delight in sharing his findings, stating, "It’s a thrilling prospect for me, because I’ve long been a fan of ‘Geek Art’." He underscored the importance of beauty in mathematics, noting that "what you can classify, you can count," which reflects his approach to solving complex problems.

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• “It’s that time of the year!” — Donald Knuth

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• “I thought this would be a good way to end my Christmas lecture, because it just looks like a wonderful Christmas decoration to me.” — Donald Knuth

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• “What a feeling of power,” — Donald Knuth, on using advanced computing resources for his calculations.

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• “It wasn’t known until this year, when they finally got the censuses working, that you can actually achieve 29.” — Donald Knuth, regarding the maximum number of angles in a Knight's Tour.

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Through his engaging presentation, Knuth not only illuminated a complex mathematical problem but also inspired appreciation for the beauty inherent in mathematics, reinforcing his legacy as a pioneer in the field.

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