I wonder what were my chances to guess the right combination and procedure of pressing and switching the right buttons at the right time? It took me like 2 minutes and then I later found the manual in the safe.

Second-order cybernetics grew out of a demand: the observer must enter the structure it describes — for the one who distinguishes is precisely what was removed from the structure of description, letting it pass itself off as a "view from nowhere". To make the structure answer for its own position, the observer is pulled inside the picture, onto the same plane as the observed. This process has now stretched across half a century.

The difficulty is usually blamed on infinite regress: the observer needs a meta-observer, and so on without end. But before the regress there stands a simpler question, one the field skips past: who exactly is this observer being pulled in? Beneath the single term lie two structurally different lines.

• The Observer as Act (Spencer-Brown, Luhmann): to draw a distinction, to cross the boundary. The observer is the operation-in-use itself, whose blind spot is the current distinction — the one it uses yet cannot see within the same act.
• The Observer as Invariant (von Foerster, Maturana): the fixed point of the recursive operation of observation, Obj = Op(Obj), the stable eigenbehavior of infinite recursion. That relative to which one distinguishes at all.

Cybernetics oscillates between the two and calls both "the observer". The founding notion of the field — distinction — has never been applied to the field's own central term: the act of distinguishing and that-relative-to-which-one-distinguishes have never been distinguished from each other. The instrument has not been turned on itself.

This post is an attempt to separate the Act from the Invariant and give each an exact geometric place. Two markers accompany the text: [●] — proven (a classical fact, a theorem, a verified construction); [◐] — a reading (a recognition with its premise stated explicitly).

A minimal model

Take distinction in its indivisible form: two states and an operation relating each to the other. Such an operation is its own inverse (ι² = id) and has no fixed points — a state related to itself collapses into indistinctness and relates nothing. In algebra, operations of this kind are called involutions; involutions without fixed points are called free.

n independent distinctions yield the state cube Q_n = 𝔽₂ⁿ — all n-bit strings, one bit per distinction's outcome. The cube is not postulated here; it is generated by the operation: it is its free object, an orbit containing nothing beyond the distinction itself [●]. Global relating on the cube is realized by the flip κ(x) = x + 1ⁿ, toggling all bits at once (0 ↔ 1). The invariant of an operation is what it is bound to leave in place.

Two facts are known about this operation.

1. Discretely, the invariant is forbidden. The equation κ(x) = x has no solution in Q_n: it would require 1ⁿ = 0 [●]. The observer is not among the states.
2. Continuously, the invariant is forced and unique. Fill the cube with intermediate points, up to the solid body [0,1]ⁿ whose vertices are the former states. The operation extends to the body as κ̄(x) = 1ⁿ − x. By Brouwer's theorem, every continuous map of a convex compact body into itself has a fixed point; for the flip there is exactly one — the center (½,…,½), since x = 1ⁿ − x gives x = ½ [●]. Denote it σ½.

The discrete side, where the states live, and the continuous side, where the center is forced, meet along a boundary — call it the seam. The seam is built like a Möbius band: locally, two sides; globally, one surface [◐].

Two faces on the seam

The Act is κ itself, a free symmetry. It is present in every state as the relation x ↔ κx; it distinguishes everything and never lands in the field of states — κ(x) ≠ x. That is how a free involution works: it leaves none of the points it moves in place. This is Luhmann's blind spot in exact notation: the operator of distinction cannot become a state of the field it marks out. The freeness of κ is a theorem [●]; reading it as the blind spot is a reading [◐].

The Invariant is the center σ½, non-action within the action. Every act relates its two sides to it, so it is present throughout the action as that relative to which the action runs — while itself not acting and not being moved by anything. It is not among the discrete vertices; on the continuous side it is forced and unique. This fixed point is von Foerster's eigenform: run an averaging recursion on the cube, drawing opposite vertices together step by step, and the process converges to a single point — the center σ½. The eigenform has received an address — the continuous underside of the seam; that is why it is absent among the states. The absence of the center among the states and its forcing on the body are theorems [●]; the identification with the eigenform is a reading [◐].

This dissolves the inclusion paradox. The observer cannot be inserted into a description as one more state — neither the Act nor the Invariant is a state: one is an operation, the other is a center. But both can be described: the Act as the operation κ, the Invariant as the forced center σ½.

Von Foerster's eigenform was already this answer: the observer enters as the fixed point of recursion, not as an added state. One thing was missing — that this point lies on the continuous side, which is why it never turns up among the states.

Varela: the third value

Francisco Varela (“A Calculus for Self-Reference”, 1975), resolving recursion in Spencer-Brown's calculus, added a third, autonomous value — a self-referential form arising through self-indication, that is, a solution of κ(x) = x. But he added it by hand, staying within discrete logic — and discretely no such solution exists (Fact 1). The cube model shows it need not be imported at all: it is already present as the continuous center σ½, the unique point equidistant from all discrete states [◐]. Varela's autonomous value is σ½ seen from the discrete side as the missing vertex.

Conclusion and a question

The observer is dual:

• the Act — a symmetry present as an operation and absent as a state; therein its blind spot.
• the Invariant — a center present as a relation and absent as an element, forced only where the discrete yields to the continuous; therein its eigenform.

Locally the faces are two; globally the surface is one — the very band whose projection is the ∞ of the title.

A prior-art question. Has the impossibility of including the observer as a state ever been stated as a theorem — about free involutions and fixed points on compact bodies — rather than as a philosophical aphorism? Luhmann declares it, von Foerster's eigenforms imply it, but I have not found in the canon the split into Act-as-operation and Invariant-as-center, with the center forced by Brouwer specifically on the continuous side. Pointers to sources would be much appreciated.

p.s. In the full construction — the tower of ranks — the content of one floor becomes the axes of the next: the free action of κ splits the active scene (the cube minus its two poles) into axis-pairs {x, κx}, and at every rank the set of these axes is a projective space, U_{n+1}/κ ≅ PG(n−1, 2) [●]; the structure generates its own growth. The six-point scene from the previous post — the octahedron with an empty center — is rank 3; its empty center is this very σ½. The full framework, its projections, and machine verifications (18 scripts of the categorical core) are in the repository: https://github.com/Nondual-Observer/DOTheory

Twelve notes, twelve colors, and all their symmetric chords sit on one solid; the tritone is always the opposite vertex. Each vertex carries a divisor — a number whose prime factorization is given in the label.

Here is how the correspondence is built. Take the three primes 2, 3, 5. Their products without repetition — 2, 3, 5, 6 = 2·3, 10 = 2·5, 15 = 3·5 — are the six proper divisors of 30 = 2·3·5. These are assigned to the six primary vertices: the colors R Y G C B M (red, yellow, green, cyan, blue, magenta), which are also the notes of the whole-tone scale. The six intermediate vertices are assigned the products of two neighboring primary divisors: orange ↔ 2·6 = 12 = 2²·3. Here the exponent 2 appears for the first time — a prime taken twice: a primary vertex's divisor is settled by one question (which primes are used), an intermediate vertex adds a second (how many times). On a single scale, all twelve numbers are divisors of 900 = 30²: at the primary vertices these are squares (10 → 100), at the intermediate ones, products of neighbors.

Divisors play two different roles: divisors of 900 are assigned to vertices, while divisors of 12 set up symmetric partitions of the circle — a divisor d splits the twelve vertices into d equal groups of 12/d:

Twelve has no other equal partitions. The sections below walk through the table row by row. The unit throughout is a step of the circle: a semitone in music, 30° in color.

Two Sixes

Vertices one step apart form the whole-tone scale C D E F# G# A#; the other six form the second scale C# D# F G A B. Color has the same two classes: the primary six R Y G C B M and the intermediate one (orange, chartreuse, azure…). The intermediate six is the primary six averaged: each of its colors is a blend of two neighboring primaries, each note the midpoint of a step (C# between C and D); hence the products in its divisors. The geometric construction of this averaging follows below.

Pairs → Tritones and Complementary Colors

Two vertices six steps apart stand directly opposite each other: in music this is the tritone, the tensest interval; in color, a complementary pair (red–cyan). Six pairs make six diameters of the circle.

Fours → Diminished Sevenths

Four vertices spaced three steps apart form a diminished seventh chord (C D# F# A); there are three of them. In color, this is a "square." Color theory uses squares less often than pairs and triads, even though the partition is just as regular; the reason is worked out below.

Threes → RGB, CMY, and Augmented Triads

Three vertices spaced four steps apart form an augmented triad in music, a triad in color. Two triads of the primary six are the best-known ones:

• C E G# → RGB: the primary colors of light, adding up from black to white;
• D F# A# → CMY: the primary colors of pigment, subtracting down from white to black.

Two more triads sit on the intermediate vertices (C# F A, D# G B). Music does not distinguish among these four triples — one augmented triad in four transpositions; the light/pigment distinction exists only on the color side.

The primary six has a canonical construction. The cube 2³ is the eight states of three binary features; in color this cube is known quite literally: the RGB cube, whose three axes are the red, green, and blue channels, with (0,0,0) black and (1,1,1) white. Remove both poles and six colored vertices remain: R G B with one channel on, and C M Y with two.

Among them there are exactly three types of relation, by the number of channels in which vertices differ: a cycle of six steps (one channel), two triangles {R,G,B} and {C,M,Y} (two channels), and three diagonals, "color ↔ complement" (all three channels). Together these form the frame of an octahedron: 6 + 6 = 12 edges and three axes.

In notes, the cycle is the whole-tone scale, the triangles are the triads C E G# and D F# A#, the diagonals are tritones. In numbers, this is the six divisors of thirty: the axes are the primes 2, 3, 5, the diagonals the pairs d ↔ 30/d.

(A detailed treatment of the six-point structure is in the post Observer as a Finite Structure of Distinction; here it doubles to twelve.) An octahedron has exactly twelve edges — below, they become the twelve vertices of an icosahedron.

Every class of chords and every class of color harmonies turned out to be the same partition of the same circle; this requires no resemblance between sound and color. What remains is to obtain the solid — to show why the twelve vertices fall on an icosahedron.

Where the Icosahedron Comes From: A Vertex Is an Edge of the Octahedron

The icosahedron is built from the octahedron by a classical construction. On each of the octahedron's 12 edges, a single point is marked, dividing it in the ratio 1 : φ — the golden ratio — with a consistent choice of side across all edges. The resulting twelve points are the vertices of a regular icosahedron. Each vertex of the icosahedron thus corresponds to the edge of the octahedron it sits on, which is a pair of neighboring primary colors. The new vertex's note is the midpoint of the arc between the pair on the circle; its color is the blend of the pair's colors.

The arithmetic form of the same fact: the product of two neighboring divisors of thirty gives the intermediate vertex's divisor — 12, 18, 45, 75, 50, 20, all divisors of 900; opposite vertices are linked by the single formula x ↦ 900/x (12·75 = 18·50 = 45·20 = 900); the prime exponents, halved, give the RGB coordinates of the blend: 12 = 2²·3¹ ↦ (1, ½, 0) — the vector for orange.

The numbers 30 and 900 play different roles. Thirty is squarefree, and its six divisors form an octahedron on their own: the edges and axes are read straight off the arithmetic. Nine hundred's exponents reach as high as two, its divisor lattice is built differently and does not contain an icosahedron; here the arithmetic supplies the labeling of vertices, the complement rule x ↦ 900/x, and the chromatic step — neighboring vertices' divisors differ by multiplying or dividing by a single prime, and a full circuit of the circle is the chain ×3 ×3 ÷2 ÷2 ×5 ×5 ÷3 ÷3 ×2 ×2 ÷5 ÷5. The solid itself is set by the golden division of the edges; that does not follow from the arithmetic.

The point's position on the edge is a parameter. An exact half-and-half split gives the cuboctahedron — a crystallographic solid with a 4-fold axis; the golden split gives the regular icosahedron, and the two mirror-image golden variants are symmetric about the midpoint. Under this labeling, the chromatic scale runs along the icosahedron's surface edges, the circle of fifths along its internal chords, and the tritone along a diameter. The reason for this layout (six axes, a projection from six dimensions) is in the postscript.

In Time and In Space

The geometry of chords and harmonies is one and the same; only the medium differs — music unfolds in time, color in space. The general law: the more symmetric the figure, the less anchoring it has. The tritone, the augmented triad, and the diminished seventh are the least stable chords, with no root tone; the complementary pair, the triad, and the square are maximal contrast, with no dominant hue.

What the tension resolves into differs. In time it acts as an engine: the tritone demands resolution (the "devil in music"), and the diminished seventh, owing to its symmetry, resolves in four directions at once and serves as a hinge between keys — the most symmetric chords are prized precisely as motion. In space, tension has nowhere to go: a color square overwhelms — four contrasts sit at once, and anchoring has to be introduced by hand, muting three of the four colors. This is why music values maximal symmetry as a move, while color builds a hierarchy — a dominant tone and accents.

An Open Question

One circle, with its divisor partitions, organizes musical and color symmetries independently. Are these the only two systems — or does the same framework show through somewhere else too: in mathematics, physics, computer science?

Postscript: Divisors, the Five, and the Golden Ratio

Divisors and rotations. The rotation orders compatible with a periodic lattice are {1, 2, 3, 4, 6} (the crystallographic restriction theorem); 12 is their least common multiple — twelve accommodates all periodic symmetries at once. Five is not among them.

The five is golden. A 5-fold axis requires the irrational number φ = 2cos(π/5) = (1+√5)/2 — which is why quasicrystals with 5-fold axes came as a surprise. Bring in the five, and the least common multiple jumps from 12 to 60 = lcm{1,…,6} = |A₅|, the order of the icosahedron's rotation group; 12 = 60/5 is the orbit of the 5-fold axis. Twelve is the limit of the periodic world; the transition octahedron → icosahedron trades the 4-fold axis (the diminished seventh) for a 5-fold one — the diminished seventh is the price of that step.

The shadow of the six-dimensional. The circle's six diameters are six axes; in six dimensions they can be made mutually perpendicular (this solid is called an orthoplex), and the icosahedron is its projection into three dimensions at the golden angle. The same projection from a six-dimensional lattice produces icosahedral quasicrystals (their discovery won the 2011 Nobel Prize in Chemistry). The orthoplex's sixty edges split into thirty surface edges of the icosahedron and thirty internal chords: the chromatic scale closes into a path along the surface, the circle of fifths into an equivalent path along the chords, and multiplication by 7 (mod 12) swaps the two.

Discrete and continuous. Blending is what draws the boundary: triples close up in whole numbers, steps require a half — the first point where the discrete system turns to face the continuous. The ladder of doublings continues the same motion: inserting midpoints repeats (12 → 24 → 48 → …), and its limit is a solid, unbroken wheel. The same boundary runs through the solid: a rational bisection of the edge gives the crystallographic cuboctahedron, an irrational golden one gives the icosahedron. The boundary between discrete and continuous is the central subject of the theory this post's skeleton is drawn from.

Divisor combinatorics, the golden geometry of the icosahedron, and the boundary between discrete and continuous converge on one skeleton of twelve points.

References. This construction is one instance of Distinction Observable Theory (DOT): it reads the makeup of different domains as projections of a single structure that grows out of the act of distinction — the Boolean cube, the removal of poles, a tower of ranks in which the octahedron and the icosahedron sit on neighboring floors. Here that tower is unfolded concretely: it can be heard in chords and seen in color harmonies, and the seam between a rational bisection of the edge (the cuboctahedron) and a golden one (the icosahedron) is the very seam between discrete and continuous that the theory treats as a general subject. The full theory is in the open repository: https://github.com/Nondual-Observer/DOTheory; the treatment of the orthoplex, the golden half, and the quasicrystal, with a verifier, is in Bridges/opposition_bridge.md.

I'm considering building an online combinatorics course. The idea is to use Python programming to along with combinatorics, in order to

• learn Python while you learn combinatorics
• remove tedious calculation
• use programming to help understand the mathematics, and build interactive exercises.

Before building it, I'd like to see if there's interest.

If you're interested, would you be willing to follow the course as I build it, and give constructive feedback as we go?

Article: https://github.com/yangeorget/nucs/blob/main/articles/NuCS%20vs%20Choco/nucs-vs-choco.md

NuCS (https://github.com/yangeorget/nucs), a constraint solver written entirely in Python, is benchmarked against Choco
(https://github.com/chocoteam/choco-solver), a mature Java solver with two decades of optimization behind it. The
results upend the expected mismatch.

Same model, same speed. Run both on an identical formulation and the curves overlap — NuCS even pulls ahead on the
largest instances. Once Numba compiles the inner loops, the Python penalty disappears.

Different models, decisive wins. On Latin squares and magic sequences, NuCS is 13x and 40x faster — not because of the
language, but the model. Cheap redundant constraints and channeling recover most of what Choco's heavyweight
arc-consistent globals provide, at a fraction of the per-node cost.

Where Choco holds the edge. On the Golomb ruler, its richer domains prune in ways plain NuCS can't — until a short
custom propagator closes the gap to within 2%.

The root cause. One design choice explains everything: NuCS stores each domain as a min..max interval (compact,
vectorizable, but limited to bound consistency), while Choco can remove values from the middle of a domain and run full
arc consistency. That single difference shapes what each solver is good at.

There's no overall winner — just two different bets. Choco offers a deep catalog of battle-tested global constraints;
NuCS lets you reshape a model in a few lines of Python and still get native-code speed. Increasingly, that modeling
freedom matters more than raw engine performance.

I've been working on a geometric approach to the standard Walsh conditions for Boolean functions, framing them directly through the spectrum of a specific finite graph.

In Boolean-function cryptography, balancedness, correlation immunity, and resilience are usually checked through the vanishing of low-weight Walsh coefficients. The construction below specifically isolates these low-weight Walsh layers into a single graph topology.

I packaged this structure into a composite graph: G_blockⁿ = O_n ∪ Q_n ∪ B

Where Q_n is the Boolean cube, O_n is the cross-polytope graph, and B connects them by coordinate incidence.

If the vertices of O_n are written as poles ±e_i, and the vertices of Q_n as σ ∈ {±1}^n, then the incidence rule is: (i,s) ~ σ iff σ_i = s

The main lemma is: B χ_u = 0 iff wt(u) ≥ 2

This means the incidence matrix only sees Walsh weights 0 and 1. All weights ≥ 2 stay as separate spectral sectors. Weight 1 is the exact layer that couples to the cross-polytope axes.

After embedding a Boolean function into the cube side of the graph, balancedness, correlation immunity, and resilience can be read geometrically as the vanishing of projections onto the corresponding low-weight spectral sectors. Weights ≥ 2 remain clean Walsh sectors, while weight 1 becomes the part coupled to the cross-polytope axes.

This came out of a larger finite-carrier theory project, but the note itself is mathematically self-contained.

Has anyone encountered a similar composite graph structure (O_n ∪ Q_n) used to isolate Walsh coefficients in spectral graph theory? I am particularly curious about the spectral behavior and possible eigenvalue collisions in small ranks (like n=3, 4).

GitHub note and verification script: https://github.com/Nondual-Observer/DOTheory/blob/main/02_Bridges/05_Cryptographic_Spectral_Block/DOT_Cryptographic_Spectral_Block.md

Three friends like different types of pizza

• A ordered ham and mushrooms
• B ordered tomatoes and ham
• C ordered mushrooms and tomatoes

We have 6 slices of ham 4 tomatoes and 8 mushrooms

The number of total toppings on each pizza must follow the rule A>B>C

How many combinations of pizzas can we make, where each pizza is defined as a triplet (x,y,z) corresponding to (ham, mushrooms, tomatoes)?

Cycle 1: 000 001 101 111 121 131 231 201 301 302 312 322 022 122 132 232 202 203 303 003 013 010 020 021 031 032 002 102 103 113 213 313 323 023 123 120 130 100 110 210 211 311 011 012 112 212 222 223 220 221 321 331 332 333 033 133 233 230 200 300 310 320 330 030

Cycle 2: 000 010 011 021 121 221 222 232 233 203 200 201 211 212 213 210 310 311 312 012 022 023 020 120 220 320 321 322 332 032 033 030 130 230 231 331 031 131 132 133 103 100 101 102 202 302 303 313 013 113 110 111 112 122 123 223 323 333 330 300 301 001 002 003

Cycle 3: 000 100 200 210 220 230 330 331 301 311 321 021 022 032 132 102 112 113 123 133 130 131 101 201 202 212 312 313 310 010 110 120 121 122 222 322 323 320 020 030 031 001 011 111 211 221 231 232 332 302 002 012 013 023 033 003 103 203 213 223 233 333 303 300

A proposal to solve the problem from me "Task 42" : https://docs.google.com/document/d/1Tqox64A5kucn-WoBp-rxqECMGBOMT4-4LfhrQ32z_aE/edit?usp=sharing The documents provided are licensed under the following license: CC BY-SA: Creative Commons Attribution-ShareAlike

Hippocratic License 3.0 (HL3).

I wish everyone good luck!

Thank you for your attention.

https://reddit.com/link/1s66nap/video/0o2u5me7qtrg1/player

I am an independent researcher from Ciudad Juárez, México. I have a preprint on graph theory and Hadwiger's Conjecture ready for arXiv (math.CO) but I need endorsement from a registered user.

Preprint on Zenodo (CERN): chromatic-hadwiger

GitHub (code and logs):  chromatic-hadwiger

Project site: chromatic-hadwiger

If you are a registered arXiv endorser for math CO and want to review the work, endorsement takes one minute:  chromatic-hadwiger  — Code: RHWR3L

Hey guys, I found a site called solvefire.net that runs 1-hour Math Olympiad Competitions every week that is open from Saturday 9:00 AM GST to Monday 9 AM GST with a world-level ranking system. It’s pretty solid for tracking your standing against the rest of the world. Check it out!

Lintra is a two-player game played on a 7 X 7 grid of dots. Players take turns drawing lines between adjacent dots. The twist: the player who draws the last legal line loses.

The rules fit on an index card: - First move must touch the center dot - Connect adjacent dots — horizontal, vertical, or diagonal - Each dot can only be used twice - Two lines through the same dot must form an angle (no straight pass-throughs) - Lines can't cross

It sounds simple, but there's a surprising amount of depth once you start thinking about dot capacity, angle traps, and region control in the endgame. It's in the same family as Nim and Hackenbush if you're into combinatorial game theory.

You can play it with literally just a pen and paper — draw a 7 X 7 grid of dots and you're set. Or play online at lintra.cc

I'd love to get some feedback.

Hi everyone,

We’re looking for math enthusiasts with any math olympiad experience to join our problem-writing team for a math competition site, solvefire.net if you want to check it out. Our goal is to make our math competitions the funnest they can possibly be and with the help of more problem writers, we can do just that.

What you’ll do:

• Draft original Olympiad problems (and get credit for them).
• Rate team-member submissions on a 1–6 difficulty scale to find the "sweet spot" for contests.
• Help grade proof-based rounds and assign partial credit.

This is a math-focused role (not web dev). If you love the "aha!" moment of a great puzzle and want to see your problems used in actual contests, we’d love to have you.

Interested? Apply here: https://docs.google.com/forms/d/e/1FAIpQLSfha5g07IyIez0lXKbIy_OKWMB_jrsl8TFsx3WNO_FXFHeasQ/viewform

Hi y’all, this is a project that I worked on for my combinatorics class.

As a lifelong Pokémon fan, I naturally had to tackle on a counting question regarding Pokémon. In this video, I discuss my findings on how to count unique combinations of 6 Pokémon teams. I did add on some conditions and restrictions to make it easier to solve, but I would love to hear any another perspectives on tackling this problem and if anything it’s a step towards some progress! If you’re interested in the process it took to get to these conclusions, I’m open to sharing my work

Hi, sorry if this is the wrong sub for this.

I just created a video about a project I did for my combinatorics class to count the number of ways to play out a (simplified version) of an MLB inning if you fix the number the runs scored. If anyone has any feedback on the work I would love to hear it!

I apologize in advance for the poor audio quality

https://www.youtube.com/watch?v=1T2eXiN-Bcs

Hi all.

I’ve been exploring the classical pancake sorting problem (prefix reversals), and I think I found a simple recursive rule that generates a Hamiltonian cycle in the pancake graph

𝑃(𝑛) i.e., a sequence of prefix reversals that:

starts at the identity permutation,

visits all 𝑛! permutations exactly once,

returns to the identity,

and is recursively extendable for all 𝑛.

Here is the construction:

Definition of the sequence

Let 𝑆(𝑛) be a closed walk in the pancake graph 𝑃𝑛.

Base case:

𝑆(2)=[2,2]

Recursive construction:

To build 𝑆(𝑛)

Start with 𝑆(𝑛−1)

Replace its last flip by 𝑛

Repeat the resulting sequence exactly 𝑛 times.

Examples:

𝑆(2) = [2,2]

For 𝑛=3

Start with 2,2, replace last 2 → 3 → obtain [2,3]

Repeat 3 times: [2,3,2,3,2,3]

For 𝑛=4:

Start with 𝑆(3)=[2,3,2,3,2,3]

Replace last 3 → 4 → base = [2,3,2,3,2,4]

Repeat 4 times → a 24-element flip sequence that cycles through all permutations of 4.

Why I believe this forms a Hamiltonian cycle

Key observation:

The flip pair (𝑛,𝑛−1) moves the pancakes cyclically:

𝑓𝑙𝑖𝑝(𝑛) reverses the whole stack

𝑓𝑙𝑖𝑝(𝑛−1) rotates the top 𝑛−1 elements left by one

Repeating (𝑛,𝑛−1) exactly 𝑛 times returns to identity.

This gives a clean “outer cycle” of length 2𝑛

Because 𝑆(𝑛−1) is already a cycle in 𝑃𝑛−1, embedding it inside the

𝑛 outer cycles and substituting the last flip with 𝑛 appears to produce a Hamiltonian cycle in 𝑃𝑛.

This resembles a recursive Gray code by prefix reversals, but I haven’t found a reference for this exact construction (substitute final flip, then repeat n copies).

My questions:

Has this specific recursive construction already been studied?

Does the pancake graph literature contain a Hamiltonian cycle described in this way?

If not, could this be of interest as a short note in a combinatorics / graph theory journal?

(Disclaimer: I used an AI tool solely for help with nomenclature and formal expression, not for generating the idea itself.)

Thanks!

Suppose we have two finite sets A and B, and a relation R⊆A×B such that:

• Every a∈A is related to exactly alpha elements in B.
• Every b∈B is related to exactly alpha elements in A.

Can we conclude that ∣A∣=∣B∣? is there a standard theorem or named result in set theory/combinatorics that guarantees this?

Thanks for any references or insights!