Donald Knuth’s 2025 Christmas Lecture: The Knight’s Tours

“It’s that clip of nan year!” publication nan email from Stanford Online. As he approaches his 88th birthday, Donald Knuth returned successful early December for his special once-a-year “Christmas” lecture — a contented he’s been honoring for over 30 years.

Stanford’s beloved professor emeritus reminded his assemblage he’s still penning “The Art of Computer Programming” — a legendary book he’s been moving connected for complete 63 years (releasing it successful installments).

But this year, he took a detour of sorts: The revered mathematics and algorithms master promised to talk “recent breakthroughs successful 1 of nan astir venerable chart mentation questions that has fascinated group for much than 1,200 years: ‘Can a knight screen each cells of a chessboard without ever visiting nan aforesaid spot twice?'”

Yet wrong that question’s dazzling permutations, Knuth delivered an important life instruction — that done each nan mathematics and machine science, what he’s searching for is beauty. Shuffling cheerfully done his printouts, Knuth showed disconnected his favourite solutions to nan puzzle, for illustration a postulation of treasured snowflakes.

And pinch warmth and wisdom, he delivered a existent vacation moment.

Art and Adventures

Speaking astatine Stanford’s Nvidia Auditorium, nan speech opened pinch an announcement — that recordings of past lectures person been precocious restored! They’ve besides created a playlist pinch 26 past Christmas lectures — each nan measurement backmost to nan precocious 1990s — while Knuth’s ain webpage links to different collections of his videos arsenic acold backmost arsenic 1981.

But looking backmost complete 2025, Knuth told his assemblage playfully that “I gotta move accelerated tonight, because there’s — I conscionable had excessively galore adventures this year.” For 1 thing, Knuth’s “news” page notes that successful June, he celebrated his 64th wedding anniversary.

In April, he’d besides helped observe nan expansive reopening of nan machine subject section astatine his alma mater, Case Western University successful Cleveland. When they’d asked Knuth really they should decorate its walls, Knuth had suggested “Knight’s Tours” — those patterns formed by tracing nan way of a knight visiting each chess committee quadrate precisely once.

Knuth moreover collaborated pinch his alma mater’s creation team, and successful a new 2,200-word page connected his website, explains nan mathematics visualized connected each wall. Knuth’s website calls it “a thrilling imaginable for me, because I’ve agelong been a instrumentality of ‘Geek Art'” — a conception Knuth explained successful his Christmas lecture. “It’s a benignant of right-brain/left-brain thing. You cognize pinch 1 half of your encephalon that it’s nice, pinch nan different half you say, ‘This is conscionable a gorgeous design.'”

They’re described successful item successful “The Art of Computer Programming.” And he’d stock examples successful this Christmas lecture.

‘Sheer Beauty’

Knuth said his fondness for Knight’s Tours started successful 1973, and much than half a period later, he’d yet recovered his aged notes. “I was intrigued astatine that clip by an unsolved problem,” he told his assemblage — smiling astatine nan truth that he still hadn’t solved it, though nan mobility was first posed successful 1891.

But Knuth besides realized that each two-move operation successful a Knight’s Tour will shape an perspective aliases “wedge” — and it turns retired that identifying nan wedges formed erstwhile landing connected nan 4 mediate squares “gives maine an easy measurement to disagreement each Knight’s Tours into astir one-eighth arsenic many.” (Since each solution could beryllium rotated successful 1 of 4 directions, aliases flipped to its reflector image.)

There’s an important instruction here: What you tin classify, you tin count.

Here successful nan 21st century, Knuth had written a programme to return these “censuses” — to cipher really galore full solutions beryllium (when fixed circumstantial sets of wedge shapes for those 4 mediate squares). His book’s latest Pre-Fascicles (published draft) specifications what he says are nan “fun” information structures he used, which allowed him to yet reply that fateful mobility first posed successful 1891: How galore solutions person 16 moves for each of nan 4 imaginable mathematical slopes for nan knight’s moves?

The reply is: 103,361,771,080.

“It wasn’t really excessively difficult to find those. Just it’s not that easy to do by hand!”

Later, Knuth says he hears from “all kinds of group who are entranced by nan taxable … It’s partially conscionable — I don’t know, nan sheer beauty of immoderate of these tours, to look at, to watch ’em. It’s for illustration listening to immoderate of your favourite euphony — it’s rhyming successful immoderate way.”

During a Q&A session, Knuth tells nan assemblage that 1 mathematician moreover created a Knight’s Tour for a three-dimensional chessboard (with 4 squares connected each side). “It has beautiful symmetry.”

And soon he’s displaying an moreover much awesome consequence — nan full number of each imaginable solutions to nan Knight’s Tour problem: 13,267,364,410,532.

“This is nan number that I thought I’d ne'er cognize nan reply to erstwhile I was an undergrad.” (The number was really first calculated successful 1997 by Australian mathematician Brendan McKay.)

Fond Figures

It’s each described successful “The Art of Computer Programming: Pre-Fascicle 8a (Hamiltonian Paths and Cycles).” But that’s conscionable nan beginning. If you tie 2 lines connecting 3 squares a knight lands on, of people they’ll shape an angle, and “A batch of group person been competing pinch each different to spot what Knight’s Tour has nan astir 37-degree angles arsenic you march along!

“It wasn’t known until this year, erstwhile they yet sewage nan censuses working, that you tin really execute 29.”

Of each these 13 cardinal tours, there’s only 136 that person reached this 29.

In fact, it turns retired that for each imaginable angle, we’ve now calculated nan maximum number that tin look successful a solution — and Knuth has a fond thought for each one.

• Right angles? “Up until this year, nan best-known was 38. But lo and behold, nan census-taker recovered a measurement to do it pinch 39.”
• Straight lines? “This is benignant of astonishing because a feline successful Romania really recovered nan optimum of this 1 already successful 1932.” The maximum number is 19, and location are only 112 solutions.
• There are 56 solutions to nan Knight’s Tour puzzle that usage 42 acute angles. And if you’re trying to avoid utilizing acute angles, location are 28,000 solutions.
• What astir obtuse angles? The maximum number is 47. “The large astonishment was that nary matter what circuit you have, you’ve sewage to person astatine slightest 4 obtuse angles successful it. You can’t debar them altogether.” And that four-angle solution is unique. “Of nan 13,267,364,410,532, there’s precisely one of them. And I besides hap to deliberation this is 1 of nan astir beautiful Knight’s Tours you’ll see.”

There’s a beauty to math, and Knuth delightedly showed solutions filled pinch straight-line angles, aliases pinch intricate symmetrical wedge patterns.

But location are besides different tantalizing angles formed successful nan solutions — successful nan lines showing wherever nan knights crossed their ain path. Knuth shows a sketch from a Belgian mathematician pinch conscionable 69 way crossings.

Knuth himself had really discovered a circuit pinch 126 different intersections “years ago” while he was searching for symmetrical solutions. (It was only years later that he learned it was unsocial — nan only 126-crossing solution, retired of each 13,267,364,410,532.)

He shows much imaginable solutions — including 1 pinch nan fewest imaginable perpendicular intersections, and pinch nan most.

The reply location would beryllium “all of them.” There’s 1 64-move solution successful which each move forms portion of a perpendicular intersection.

Let There Be Light!

But nan astir analyzable census of each was “a problem that I had been reasoning astir for 30 years.” And it turned retired to beryllium a expansive escapade successful some mathematics and machine science.

Math includes nan conception of a “winding number” — nan number of times a constituent is afloat circled by a curving statement — and it’s often visualized pinch achromatic for moreover numbers and achromatic for odd. Knuth’s friend George Jelliss made a beautiful observation: “We tin picture immoderate Knight’s Tour by this black-and-white pattern.” The Pre-Fascicles of Knuth’s book see immoderate awesome examples:

So what’s nan darkest imaginable circuit — and what’s nan lightest?

It would’ve taken Knuth 8 months to cipher it connected his location computer. But fortunately, a Stanford workfellow loaned him a amended setup — 26 machines, boasting a full of 832 cores. There were 2 16-core CPUs connected each machine.

“What a emotion of power,” he said, drafting a laughter of statement from his audience. “If you tin imagine. For 3 days, I was moving complete 800 jobs astatine once!”

A ‘Whirling’ Finish

And what if that patient touring knight is ever walking counterclockwise astir nan halfway so, arsenic Knuth puts it, “He ne'er backs up!”

It’s imaginable — location are 1,120 ways connected an 8 x 8 chessboard. But Knuth shows disconnected nan patterns. “When you look astatine these tours, they disagreement into coils,” pinch each complete circuit yet crossing complete a “plumb line.”

Knuth tried to make a shape pinch arsenic galore coils arsenic location are squares to a side, collaborating pinch Bulgarian mathematician Nikolay Beluhov. Beluhov yet recovered specified a solution, connected a 12 x 12 chessboard. And past together, nan 2 created a genuinely breathtaking diagram. “We came up pinch this building that shows astatine slightest that for each N > 24 that are multiples of 4, location is simply a ‘whirling’ Knight’s Tour pinch N coils.”

“But past I said, ‘Well, what astir trying to get 1 that has symmetry nether rotation of 90 degrees?'”

And past for a expansive finale, Knuth puts up his last slide. If these were a postulation of snowflakes, this mightiness beryllium his prized possession. Not only does it person nan aforesaid number of coils arsenic squares connected its side. “This is an 18 x 18 ‘whirling’ Knight’s Tour, that if you rotate it 90 degrees, it’s nan aforesaid tour.”

And location it was — a mathematical beauty, and a ocular beauty. “I thought this would beryllium a bully measurement to extremity my Christmas lecture, because it conscionable looks for illustration a awesome Christmas decoration to me.”

And nan assemblage applauded warmly.

Previous Donald Christmas Lectures

Donald Knuth’s Christmas Lectures are an yearly contented astatine Stanford University. Each twelvemonth successful early December, nan renowned machine scientist and writer of “The Art of Computer Programming” delivers a speech connected a assortment of topics related to machine subject and mathematics, to nan delight of students and greybeards alike. These talks are captured connected YouTube. 

Donald Knuth’s 2024 Christmas Lecture: ‘Strong’ Memories

Donald Knuth’s 2023 Christmas Lecture: Making nan Cells Dance

Donald Knuth’s 2022 ‘Christmas Tree’ Lecture Is astir Trees

Donald Knuth connected Machine Learning and nan Meaning of Life (2021)

Donald Knuth’s 2019 ‘Christmas Tree Lecture’ Explores Pi successful ‘The Art of Computer Programming’

Donald Knuth’s 2018 Christmas Tree Lecture connected Dancing Links — and Organ Music

Donald Knuth’s 2017 Christmas Tree Lecture Tackles a ‘Curious Problem’ successful Combinatorial Geometry

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