51 subscribers: six points, 51 ways to draw chords, and 51 = 33 in base 16

Description:
We just reached 51 subscribers. Thank you! The 37-subscriber Short promised 38 next, the magic sum of the magic hexagon, and we skipped right past it. But we did grow 38% since that Short. Rounded: 14/37 = 0.378 378 378 …, so +37.84%. (Still finite.)

51 looks prime, but in base 16 it's 33: three sixteens and three ones. That's three seventeens, so 51 = 3 × 17. In any base, a two-digit repdigit aa is a × 11, just as the last Short's three-digit repdigits were multiples of 111 = 3 × 37.

And the 51 drawings in the cold open: put six points on a circle and join some pairs with chords, as long as no two chords touch (no crossing, no shared point). There are exactly 51 ways, counting the empty one: 1 with no chords, 15 with one, 30 with two and 5 with three. That makes 51 a Motzkin number: 1, 1, 2, 4, 9, 21, 51, 127, …

Next milestone: 64 = 8² = 4³, the first number after 1 that is both a square and a cube, and the last Short's cube (27 + 37). We'll try not to skip it.

What is what:
• Recomputed exactly in Tungsten: of the 2¹⁵ = 32,768 sets of chords among six points, exactly 51 have no two chords touching, by number of chords 1, 15, 30 and 5; the same brute force for 0 to 7 points gives 1, 1, 2, 4, 9, 21, 51, 127, equal to the recurrence M(n) = M(n − 1) + Σ M(k) M(n − 2 − k) and to Σ C(n, 2k) Cat(k), both checked to n = 20. 51 in base 16 is 3, 3; 51 is not prime (3 × 17); among bases 2 to 64 it is a two-digit repdigit only in base 16 (33) and base 50 (11). The growth 14/37 by long division (period 3: 10³ = 999 + 1, and 999 = 27 × 37). The magic hexagon's 15 lines all sum to 38. No number from 2 to 63 is both a square and a cube. Controls that must fail, and do (10 of 10): 51 prime, 51 = 32 in base 16, chords allowed to cross (76 ways), chords allowed to share a point (2,880 ways), 51 ways on seven points (127), growth exactly 38% or rounding to 37%, the magic hexagon with two cells swapped, 27 a square, 36 a cube.
• Kernel-checked in Satori (fiftyone.sat, 14 theorems, one process at a time): the drawings on six points counted by the classical recursion (point 0 alone, or joined to a point that splits the rest into inside and outside), 1, 15, 30 and 5 by number of chords, and the Motzkin numbers to seven points; 51 = 3 · 17, not prime; 51 = 3 · 16 + 3; aa = a · 11 in every base; the magic hexagon's 15 lines; 14/37 = 378/999; 64 = 8² = 4³ = 27 + 37, and no smaller square cube but 0 and 1. The milestone, subscribers = 51, rests on one named axiom, youtube_counter_is_correct_yet_again, and Satori reports it with the verdict. A control copy claiming 52 drawings is refused.
• Cited: T. Motzkin, "Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products", Bull. AMS 54 (1948); OEIS A001006 (Motzkin numbers).
• The music is our own, synthesized in Tungsten. In the cold open the chord the 37-subscriber Short landed on (110, 440, 550, 660 and 880 Hz) comes back under the first 37 drawings, and each new subscriber is a music-box note on the next harmonic of 55 Hz, 38 to 51 (2,090 to 2,805 Hz), carrying on from the earlier celebrations' 1 to 37. In the finale the six points are six notes (harmonics 8, 10, 12, 16, 20 and 24 of 55 Hz: A, C♯, E over two octaves), every chord sounds its two ends (a diameter is an octave), the five drawings with three chords play all six, over the 10-subscriber Short's chord (harmonics 1–10 of 55 Hz), landing on an A-major chord. The pelican from the channel banner rides by on six-spoked wheels.
• This celebration is still not peer reviewed.

Every celebration so far:
https://www.youtube.com/playlist?list=PLIhCurG6GvnQ

More explainers, in the openai/math playlist:
https://www.youtube.com/playlist?list=PLeyvICL-ZNKo

The channel:
https://www.youtube.com/@UnstableMaths

Tungsten, the language every computation, frame and sound is written in:
https://tungsten-lang.org

Satori, our proof checker:
https://github.com/tungsten-lang/tungsten/tree/main/bits/tungsten-satori

Pictures and music made in Tungsten; proofs checked in Satori; narration by ElevenLabs.

Independent channel, not affiliated with OpenAI.

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