The family of layered cubes [frap-0003]
✍️source

    To pass from one cube size to another, we first fix a physical model. Let \(C_N\) be an \(N\times N\times N\) array of unit cubies, with coordinates \((i,j,k)\in {1,...,N}^3\). A visible cubie is a surface cell, so at least one coordinate is \(1\) or \(N\). A cell with three boundary coordinates is a corner; with exactly two it is an edge; with exactly one it is a center. This is a description of positions, before we decide which labels a player can see.

    The resulting counts are immediate: there are \(8\) corners, \(12(N-2)\) edge cubies, and \(6(N-2)^2\) center cubies. Their sum is the number of surface cells. The count already explains why changing \(N\) is more than making a larger drawing. New edge and center positions appear, and those positions can fall into different move orbits. Bonzio, Loi and Peruzzi use this cubie decomposition in their general model [bonzio2018nxnxn, Sections 2--3]; Salkinder gives a parallel orbit and configuration analysis [salkinder2021nxnxn, Section 2].

    The abstract-to-cube handoff is made explicit in how the cube models connect. In the \(N=3\) case, the 54 facet-colored incidence positions group into corner and edge sticker fibers, with six fixed centers supplying the frame; the resulting action is the fixed-frame coordinate model used later. For larger \(N\), the new center and coupled-edge orbits prevent this from being a single ordinary-cube coordinate space.

    1. Layer turns and the fixed frame

      Choose the six coordinate directions as a fixed frame. For an axis \(a\in {x,y,z}\), a depth \(r\) with \(1\le r\le \lfloor N/2\rfloor \), and either boundary of that axis, turn the layer \(a=r\) or \(a=N+1-r\) by a quarter-turn. The turn rotates the other two coordinates and carries every sticker on a moved cubie with it. For odd \(N\), the central layer is deliberately omitted; this is the usual fixed-center convention. Thus \(N=3\) has the six outer-face turns, while \(N=4\) has the outer and adjacent inner layers.

      This choice distinguishes a layer turn from a whole-cube rotation. Whole-cube rotations change the chosen frame and are not generators of the state group here. They can be added as a separate symmetry action, or used to identify observations later. Keeping them out makes the six face directions, and hence a fixed center color scheme, part of the state description.

      A labelled state assigns a distinct name to every physical cubie and, when needed, to every sticker. A move acts by permuting these labels and their local frames. An ordinary colored observation forgets distinctions between equal-colored stickers; a marked or 'supercube' observation retains them. These are different quotients of the same labelled action. In particular, a statement that two center cubies 'look the same' is an observation claim, not a claim that the labelled pieces are interchangeable by a move.

      2. The coordinate signatures of the orbits

        The useful invariant is the distance of a coordinate from the nearest boundary. A layer turn permutes two coordinates, possibly changes their signs relative to the cube center, and leaves the multiset of these distances unchanged. It therefore preserves corners, edges and centers. On an edge, the one nonboundary coordinate gives a ring index. On a center, the two interior coordinates give an unordered pair of distances. Equal distances give a center-corner position in a concentric ring; unequal distances give a center-edge position. The central pair in an odd cube is the fixed center.

        Write \(K_e=N/2-1\) for even \(N\) and \(K_o=(N-3)/2\) for odd \(N\). The position orbits under the selected layer turns have the following sizes. For even \(N\), the coupled-edge orbits \(E_1,...,E_{K_e}\) each have \(24\) positions. For odd \(N\), there is a single-edge orbit \(E_s\) of size \(12\), together with coupled-edge orbits \(E_1,...,E_{K_o}\) of size \(24\) each. The term coupled records that the edge is away from the odd cube's middle edge position; it does not by itself choose an orientation convention.

        For even \(N\), each center-corner ring \(C_k\), \(1\le k\le K_e\), has \(24\) positions. The center-edge ring \(D_k\), \(2\le k\le K_e\), has \(48(k-1)\) positions, split into \(2(k-1)\) distinct orbits of \(24\). For odd \(N\), there are \(6\) fixed centers, center-corner rings \(C_k\) of size \(24\), and center-edge rings \(D_k\) of size \(24(2k-1)\), each for \(1\le k\le K_o\). Each odd ring is likewise a union of \(2k-1\) orbits of \(24\). Summing these rows gives \(6(N-2)^2\) centers in both parities. The formulas expose the distinction between a position orbit and an orientation coordinate: a one-sticker center has no orientation unless it is specially marked, while a coupled edge can have a convention-dependent orientation record.

        These sets are not merely a count. Face turns carry a representative edge around the four faces of its ring; an inner layer at depth \(k\), conjugated by outer turns, carries it to the other axis directions. The same construction carries a center-corner representative around its ring and a center-edge representative around every position with the same ordered distance-pair or chirality label. Hence each displayed \(24\)-position set is a move orbit, while the distance signature proves that different rows cannot mix. This is the coordinate part of the model; legality constraints on the permutations are a separate question.

        3. Small cubes make the parity split visible

          For \(N=2\), \(K_e=0\): there are only the eight corners. For \(N=3\), \(K_o=0\): the pieces are eight corners, twelve single edges and six fixed centers. This is the familiar cube, but the description explains why its centers do not move in the chosen layer family.

          For \(N=4\), \(K_e=1\): there are eight corners, one coupled-edge orbit of \(24\), and one center-corner orbit of \(24\); there are no fixed centers and no center-edge ring. For \(N=5\), \(K_o=1\): there are eight corners, twelve single edges, one coupled-edge orbit of \(24\), six fixed centers, one center-corner orbit of \(24\), and one center-edge orbit of \(24\). The latter has \(8+36+54=98\) visible cubies, the familiar Professor's Cube count. These examples show what changes with parity without claiming that the same labels or orientation coordinates are used by every source.

          The ordinary cube action is the \(N=3\) outer-layer case of this model. The next chapter can therefore state constraints for a fixed three-layer convention, while the all-\(N\) first law must still reconcile the labelled, colored and coupled-edge conventions in Bonzio--Loi--Peruzzi and Salkinder. Salkinder writes all slice rotations, including a central slice in odd size, and then treats its use through a shifted middle-slice convention; this is a genuine convention choice, not a harmless notation change. We use the fixed-center generators above and leave the translation between these move families explicit. A necessary invariant will not be promoted to a sufficiency theorem until that reconciliation is written out.

          References

            Reference. Nxnxn rubik’s cubes and god’s number [salkinder2021nxnxn]

            @misc{salkinder2021nxnxn,
 title = {nxnxn Rubik's Cubes and God's Number},
 author = {Daniel Salkinder},
 year = {2021},
 url = {https://arxiv.org/abs/2112.08602},
 primaryclass = {math.CO},
 archiveprefix = {arXiv},
 eprint = {2112.08602}
}

            Reference. On the nxnxn rubik’s cube [bonzio2018nxnxn]

            • 2018
            • Stefano Bonzio, Andrea Loi, Luisa Peruzzi
            @article{bonzio2018nxnxn,
 title = {On the nxnxn Rubik's Cube},
 author = {Bonzio, Stefano and Loi, Andrea and Peruzzi, Luisa},
 year = {2018},
 journal = {Mathematica Slovaca},
 volume = {68},
 number = {5},
 pages = {957--974},
 publisher = {De Gruyter}
}

            Backlinks

              Five encounters with a cube [frap-001C]

                Put a solved three-layer cube on a table and turn its right face once. The move is familiar, but it raises concrete puzzles: which labelled pieces moved, why can a visually plausible arrangement be impossible, and how can another person undo the turn without watching the first hand? Begin with these experiences, then introduce only the distinctions needed to explain them.

                One turn is larger than the visible face

                  The shaded strip is only the visible part of the layer: the same stickers cross the front, right, back, top, and bottom faces; viewed from above, the top-face part is the corresponding rightmost strip. A whole-cube rotation would change the viewing frame, whereas this face move keeps the six centres fixed and changes one layer. The layered-cube family makes this distinction for every \(N\); the complete three-layer cube spells it out for the six outer generators.

                  A sticker needs a home and a frame

                    A player sees colours; a proof must also track each sticker's cubie and transported local frame. How the cube models connect follows one sticker through incidence position, cubie fibre, and fixed-frame coordinate. A colour quotient forgets labels, while a larger cube adds positions, so the state space changes with the question.

                    A plausible state can still be impossible

                      Take a solved cube and twist one corner in place. Every sticker still has a cubie-sized home, and the three colours at that corner look locally consistent. But the corner orientations now have sum \(1\) modulo \(3\), whereas every face turn preserves sum \(0\). No sequence of legal turns can produce this state.

                      The complete three-layer cube derives the matching edge and permutation restrictions from the six generators. The calculation is useful at the table: it tells a solver when a scrambled-looking object was assembled illegally, and it tells a proof which state space is worth searching.

                      A human algorithm has to preserve these legal states while it borrows pieces from completed areas.

                      A formula often disturbs pieces that were already placed. The disturbance is acceptable only when the setup is undone and the completed pieces return. A working slot, a trigger, and an inverse setup make that promise visible. The complete layer-first method repeats this pattern through the cross, the first layer, the middle layer, and the last layer. Its completeness depends on the preservation and termination checks, not on the formula's familiarity.

                      Undoing a scramble checks the notation

                        The worked replay and projected lower bound fixes the viewing convention, records the ten tokens, and applies their inverse. Sticker replay returns to solved. A corner-only projected search finds no solution of depth at most two, so the evidence is the bounded statement

                        \[3\le d_{\rm HTM}\le 10.\]

                        The inverse word is an upper-bound witness; the projected search is a lower-bound witness. The computer account explains why these checks do not prove that ten is shortest. A state, a word, and a replay log are enough to test the notation before asking for a general solver.

                        Changing one ingredient changes the problem

                          Once the move, its frame, its local repair, and its replay are explicit, three variations separate the effects of enlarging the cube, changing the shape, and forgetting labels.

                          The layered-cube family develops the first variation; the incidence and facet construction in incidence, symmetry and moves develops the second; and which moves survive a quotient? gives the descent condition for the third. Framed orientations, label transport and topology, and scrambling, spectra and search then study the extra information each variation introduces or forgets. Their hypotheses stay attached to the concrete move that prompted the question.

                          A face turn can therefore be drawn, named, replayed, and checked before it is generalized. The resulting state maps and invariants support later formal certification or measured comparisons, while the examples here make clear which claim each later programme would have to establish.

                          Further connections [frap-0007]

                            The total-orientation character can expose a constraint, but what survives when we change the local frame or the dimension? Framed orientations and transfer and dimension develop the first connections: wreath product records positions and local transport, while abelianized characters detect constraints when the allowed generators preserve them [marchetti2025rubik, Section 3.1]. Finite-index transfer organizes related calculations through coset representatives [mayer2015artin, Sections 3.1--3.3]. The dimension calculation explains how a local abelianization can lose this detecting power without deciding reachability.

                            The following outlooks ask different questions about transport and functions on a move group. Each needs its own construction; neither is a prerequisite for continuing to the layered cubes.

                            Label transport and topology [frap-0014]

                              A sequence of puzzle moves may return pieces to their positions while changing their orientations. The first question is what local return map records that change; the cycle criterion computes it. A related question asks how labels move between neighboring simplices. The projectivity construction of Izmestiev and Joswig makes that question precise, and the complete unfolding records the topology of one such transport system [izmestiev2002branched, Sections 2.1--2.3.3].

                              Perspectivities along facet paths

                                Let \(K\) be a pure \(d\)-dimensional simplicial complex. If neighboring facets \(\sigma \) and \(\tau \) share a ridge, \(\sigma \) has one vertex \(v(\sigma ,\tau )\) not in \(\tau \). The perspectivity

                                \[h_{\sigma ,\tau }:V(\sigma )\longrightarrow V(\tau )\]

                                swaps that exceptional vertex with \(v(\tau ,\sigma )\) and fixes the common ridge vertices. For a facet path \(\gamma =(\sigma _0,\ldots ,\sigma _n)\), its projectivity is the composite

                                \[\langle \gamma \rangle =h_{\sigma _0,\sigma _1}\cdots h_{\sigma _{n-1},\sigma _n}.\]

                                A closed facet path based at \(\sigma _0\) gives a permutation of \(V(\sigma _0)\). The set of these permutations is the projectivity group \(\Pi (K,\sigma _0)\leq \operatorname {Sym}(V(\sigma _0))\); changing the base facet conjugates the group when the dual graph is connected. The inverse path supplies the inverse permutation, so this is transport with a genuine composition law [izmestiev2002branched, Section 2.1].

                                Coloring as a trivial-transport test

                                  A proper coloring of the vertices of \(K\) with \(d+1\) colors gives every facet a consistently labeled vertex set. Each perspectivity preserves those labels, so every projectivity is the identity. Thus proper coloring is a sufficient condition for \(\Pi (K)=1\); a nontrivial projectivity group is an obstruction to such a global labeling [izmestiev2002branched, Section 2.1].

                                  This is a useful comparison with a framed puzzle. A local frame assigns labels in one fiber, and a move transports them to another fiber. The framed transport and its total-orientation character measure the return action in the puzzle model. Projectivities do the analogous bookkeeping for simplex vertices, but the two constructions have different state spaces and move laws. No identification between their groups is being assumed.

                                  Complete unfolding as a transport cover

                                    Fix a base facet and a labeling of its vertices by \(0,\ldots ,d\). A facet reached by a path inherits a labeling by transporting along that path. Different paths can give different admissible labelings, precisely when their projectivity differs. The complete unfolding keeps these admissibly labeled facets as separate sheets and glues neighboring sheets only when their labels agree on the common ridge. It carries a canonical map back to \(K\); the base projectivity group acts by permuting these sheets. The complete unfolding itself has trivial projectivity group [izmestiev2002branched, Section 2.2, Proposition 2.2.2].

                                    The construction is therefore a topological record of transport failure: a loop in the base may close geometrically while changing the sheet label. The complete unfolding removes that ambiguity by remembering the transported labeling. This is analogous to a covering-space monodromy calculation; the branch locus is additional local-combinatorial data of the unfolding and is not inferred here from the puzzle model.

                                    The tetrahedral boundary example

                                      For the boundary complex \(\partial \Delta ^3\), the projectivity group is \(S_3\). Its complete unfolding is glued from 24 triangles. Identifying opposite sides of the triangulated hexagon gives a torus \(T^2\); the canonical map to the tetrahedral boundary, topologically \(S^2\), is a six-fold branched covering. There are four branch points on \(S^2\); each has three preimages, each of branching index two [izmestiev2002branched, Section 2.3.3, Figure 3].

                                      The example separates three pieces of data that are easy to conflate: the base triangulation, the permutation group generated by facet-path transport, and the topology of the unfolded surface. The torus is not a claim about a cube solver. It is a concrete model showing how local label transport can assemble a global branched cover.

                                      A comparison between these constructions must name the fiber, the local transport map, and the allowed paths before relating topology to puzzle orientation. The projectivity example supplies one transport vocabulary; the framed cube account supplies a different finite-state calculation. A bridge between them would require an explicit map between those data and a proof that it preserves composition, return maps, and the claimed topology.

                                      Scrambling, spectra and search coordinates [frap-0015]

                                        Random moves raise a different question from solving: when, and how quickly, does a scramble lose information about its starting state? A random walk depends on its allowed moves and their probabilities. We use the fixed-frame three-layer move group from the complete cube account, and write \(G\) for its reachable state group. A walk with step law \(\mu \) has distribution

                                        \[\Pr [X_k=g]=\mu ^{*k}(g)\]

                                        from the solved state, where convolution records the successive moves.

                                        A parity calculation before mixing claims

                                          Let \(\varepsilon :G\to \{+1,-1\}\) be the sign of the corner permutation. Each face quarter-turn or inverse is a four-cycle on corners, so \(\varepsilon =-1\); each face half-turn is a product of two transpositions, so \(\varepsilon =+1\). If the walk chooses uniformly from the twelve quarter-turns and inverse quarter-turns, then

                                          \[\varepsilon (X_k)=(-1)^k\]

                                          with probability one. Its support is therefore contained in one parity coset at every fixed time, while the uniform distribution on \(G\) gives equal mass to the two cosets. The total-variation distance from uniform is at least \(1/2\) for every \(k\). This walk cannot converge to uniform: the obstruction is periodicity, not a slow estimate of a mixing time.

                                          The usual eighteen-move HTM law gives a different exact observable. Six half-turns have sign \(+1\) and twelve quarter-turns or inverses have sign \(-1\), hence one step has

                                          \[\mathbb E[\varepsilon (X_1)]=\frac 6{18}-\frac {12}{18}=-\frac 13.\]

                                          Independence gives \(\mathbb E[\varepsilon (X_k)]=(-1/3)^k\). Since uniform parity has expectation zero, the parity marginal alone gives the lower bound

                                          \[\lVert \mathcal L(X_k)-\mathcal U_G\rVert _{\mathrm {TV}}\geq \frac 12\left (\frac 13\right )^k.\]

                                          This bound decays and does not prove convergence by itself. A lazy walk, which includes the identity with positive probability, removes this specific periodicity; a convergence statement still needs a generated reachable component and an aperiodicity argument.

                                          Averaging and finite-group spectra

                                            For a function \(f:G\to \mathbb C\), the one-step averaging operator is

                                            \[(T_\mu f)(g)=\sum _{s\in G}\mu (s)f(gs).\]

                                            Constants are fixed by \(T_\mu \). For a finite-dimensional unitary representation \(\rho \), its matrix coefficients

                                            \[f_{u,v}(g)=\langle u,\rho (g)v\rangle \]

                                            form invariant blocks for this operator: averaging a coefficient replaces \(\rho (g)\) by multiplication by the Fourier block

                                            \[\widehat \mu (\rho )=\sum _{s\in G}\mu (s)\rho (s).\]

                                            Thus the nonconstant spectrum describes which observables retain memory. If the step support generates the reachable group and every nontrivial block contracts on the relevant quotient, then these blocks decay and the walk converges to uniform on that component. The hypotheses matter: the quarter-turn walk above fails the aperiodicity test, so no spectral slogan can erase its parity obstruction.

                                            The same calculation explains why a prescribed scramble length is not a universal guarantee of randomness. A distance bound answers how far a state can be from solved; a distributional claim asks how the whole law projects onto every persistent observable. These are different certificates.

                                            Fourier coordinates for learned search

                                              Pan and Kondor use irreducible-representation matrix coefficients as a Fourier basis for learned value functions on permutation puzzles [pan2021fourier, Sections 4--4.1]. Their experiments compare learned scores that choose moves with conventional heuristic and value-based search; the paper explicitly discusses the distinction from an admissible lower bound [pan2021fourier, Section 6]. A learned score can rank promising moves while overestimating or underestimating the remaining distance. It is therefore evidence about search guidance, not a shortest-path certificate.

                                              For the cube, a safe lower bound can still be obtained from the exact projected distances in a pattern database as in computer solving and efficiency. A Fourier score may be combined with that bound as a tie-breaker or move-ordering device, but replacing the bound with the score changes the guarantee. The representation coordinates explain how a search function is organized; they do not, without an error theorem, make it admissible.

                                              The displayed parity bounds and finite-operator framework are the results established here. They do not claim a full-cube mixing time, a universal scramble length, or a reproduced machine-learning benchmark. Those would require a declared move law, reachable component, statistical or spectral analysis, and independently checked computational evidence.

                                              Which moves survive a quotient? [frap-0019]

                                                A quotient of observations is useful only when the moves still have a well-defined action on the observations. Let \(X\) be a declared sticker-state set, let \(G\) act on \(X\), and let the surjection \(q:X\twoheadrightarrow Y\) identify the observations we want to retain (equivalently, take \(Y=q(X)\)). The move \(g\in G\) preserves the fibers of \(q\) when

                                                \[q(x)=q(x')\Longrightarrow q(gx)=q(gx')\qquad (x,x'\in X).\]

                                                Exactly in this case there is a unique induced map \(\bar g:Y\to Y\) with

                                                \[q\circ g=\bar g\circ q.\]

                                                If \(g\) is a bijection and \(X\) is finite, then \(\bar g\) is surjective and therefore bijective. In an infinite setting, require the inverse move to preserve fibers as well. Thus, for the finite cube action below, the fiber-preserving bijections form a subgroup; a quotient does not automatically inherit the whole puzzle action.

                                                The antipodal sticker quotient

                                                  For the ordinary cube, use the surface-cell and frame convention in how the cube models connect and the complete three-layer cube. Represent a sticker by a pair \((p,n)\), where \(p\in \{-1,0,1\}^3\) satisfies \(\lVert p\rVert _\infty =1\) and \(n=\operatorname {sgn}(p_i)e_i\) for each nonzero coordinate \(p_i\). Thus corners, edges and face centers contribute \(8\cdot 3\), \(12\cdot 2\) and \(6\cdot 1\) pairs, respectively, for 54 stickers. The antipodal identification is the involution

                                                  \[a(p,n)=(-p,-n),\qquad q(p,n)=\{(p,n),a(p,n)\}.\]

                                                  Its sticker fibres pair opposite stickers. This is a quotient of the labelled sticker action; it is not the same as forgetting equal colors or quotienting by whole-cube rotations.

                                                  Why one face turn fails

                                                    The positive-\(z\) face turn uses

                                                    \[u(x,y,z)=(y,-x,z)\]

                                                    on every sticker whose cubie has \(z=1\), rotating both its position and its normal, and fixes the other cubies. Take a top sticker and its antipode, for example

                                                    \[x=((1,1,1),e_z),\qquad a(x)=((-1,-1,-1),-e_z).\]

                                                    The turn sends \(x\) to the sticker at \(((1,-1,1),e_z)\), while it fixes \(a(x)\). These outputs are not an antipodal pair, so \(q(ux)\ne q(u\,a(x))\). The single \(U\) move therefore does not descend to the antipodal quotient.

                                                    A synchronized opposite-layer move

                                                      The clockwise turn of the negative-\(z\) face, viewed from that face, is the inverse global rotation on \(z=-1\). Write it as \(d\); its inverse \(d^{-1}\) uses the same global rotation as \(u\). The composite \(u d^{-1}\) rotates both opposite layers by the same linear map and fixes the middle cubies. It commutes with the antipodal involution:

                                                      \[a\,(u d^{-1})=(u d^{-1})\,a.\]

                                                      Consequently it preserves every antipodal fiber and induces a bijection on \(Y\). A direct check of all 54 stickers gives the same result: \(U\) fails on the displayed pair, while the synchronized \(U D'\) move maps every antipodal pair to an antipodal pair. The surviving action is therefore an action of the fiber-preserving subgroup, not of the six ordinary face turns individually.

                                                      What the cover does not provide

                                                        A covering or quotient of the underlying shape supplies a map of shape data. It does not by itself supply a map between sticker-state spaces, a chosen fiber relation, or an equivariance proof for each legal move. Those data are separate: the explicit cube action records which stickers a face turn moves, while the quotient records which states are identified. A move descends only after the displayed fiber test has been checked.

                                                        This is the precise limit of the comparison with the complete unfolding in label transport and topology. Projectivity and branched unfolding describe transport of simplex labels; the antipodal example describes descent of a particular sticker action. They pose analogous questions about information lost by identification, but neither construction is being identified with the other. A general lifting or descent classification is outside this note.

                                                        The practical rule is short: state the state space, the quotient map and the move action before saying that a move survives identification. The antipodal calculation supplies both a negative and a positive finite example, while preserving the distinction between a shape cover and a compatible puzzle action.

                                                        How the cube models connect [frap-001A]

                                                          An abstract cube turn records incidence positions and restricts a shape symmetry to one facet. A layered-cube turn records cubies, stickers, frames and allowed layers. For the ordinary three-layer cube these are two descriptions of the same action; the map below identifies their coordinates.

                                                          From facet incidence to the 54-sticker state

                                                            The facet calculation in one ordinary cube face turn selects the facet-colored part of the incidence set. Put the cube's surface cells at \(p\in \{-1,0,1\}^3\ \text {with}\ \lVert p\rVert _\infty =1\). A visible sticker is a pair \((p,n)\), where \(n=\operatorname {sgn}(p_i)e_i\) for each nonzero coordinate \(p_i\). Thus a corner has three visible normals, an edge has two, and a face center has one:

                                                            \[8\cdot 3+12\cdot 2+6=54.\]

                                                            The vertex--facet incidences in the facet-colored set map to the pairs above by sending a cube vertex, edge, or facet to its surface-cell coordinate and its incident facet normal. Conversely, the zero-coordinate pattern of \(p\) determines the cubie (three nonzero entries for a corner, two for an edge, one for a centre), and \(n\) determines the incident facet, so this is a bijection between the 54 incidences and the 54 pairs. The cubie counts and this incidence decomposition follow the general model in [bonzio2018nxnxn, Sections 2--3]; the displayed pairing and 54-count are the calculation used here. For \(N=3\), group these pairs into eight corner and twelve edge sticker fibres, leaving the six center positions as fixed frame marks. A legal outer-face move preserves this partition and carries the labels in a corner fibre or edge fibre to the corresponding destination fibre. Write \(C_p=\{(p,\operatorname {sgn}(p_i)e_i):p_i\ne 0\}\) for a corner fibre and \(E_p\) for an edge fibre. For a corner, order its normals as \(L_p=(u_p,f_p,r_p)\), where \(u_p=\operatorname {sgn}(p_3)e_3\), \(f_p=\operatorname {sgn}(p_2)e_2\), and \(r_p=\operatorname {sgn}(p_1)e_1\). For an edge, put first the U/D normal when \(p_3\ne 0\), otherwise the F/B normal, and put the other normal second. A move sends \(C_p\) or \(E_p\) to a destination fibre. Let \(k_p\in \{0,1,2\}\) be the index of the transported U/D normal in \(L_{p'}\). If the source cubie has \(p_3=+1\), set \(x_p=k_p\); if it has \(p_3=-1\), set \(x_p=-k_p\) in \(\mathbb Z_3\). Thus each side turn gives two \(+1\) and two \(-1\) corner twists, while U/D turns give zero, matching the U/D-sticker convention in the complete three-layer cube. For an edge, its distinguished normal agrees or disagrees with the first destination label, giving \(z\in \mathbb Z_2\). This explicit extraction fixes each per-fibre bijection and the destination permutation without asserting an unstated cyclic convention. Therefore it induces exactly the fixed-frame coordinates

                                                            \[(\sigma ,x;\tau ,z)\in S_8\times (\mathbb Z_3)^8 \times S_{12}\times (\mathbb Z_2)^{12}\]

                                                            used in the complete three-layer cube. More explicitly, if a move sends a corner fibre \(C_p\) to \(C_{p'}\), set \(\sigma (p)=p'\); the destination edge fibre similarly gives \(\tau \). For the positive-\(z\) turn \(q(x,y,z)=(y,-x,z)\) from one ordinary cube face turn,

                                                            \[((1,1,1),e_z)\longmapsto ((1,-1,1),e_z),\qquad ((1,1,1),e_y)\longmapsto ((1,-1,1),e_x).\]

                                                            This exhibits both the destination corner and the transported local frame. The induced map is a change of state description for the legal action: it sends a sticker permutation to its corner and edge destination permutations together with the local frame changes. It does not assert that every permutation of the 54 positions is a legal cube move.

                                                            This also separates two kinds of forgetting. Passing from the incidence action to the displayed coordinates forgets the individual sticker addresses inside each recorded local frame. Passing from labelled cubies to an ordinary colour observation forgets equal-looking labels. The latter is the observation quotient discussed in which moves survive a quotient, and it is a separate identification of states rather than a change in these coordinates.

                                                            What changes when layers are added

                                                              For larger \(N\), the same surface-cell description retains the corner and edge fibers but adds coupled-edge and center orbits. Inner layers move those new fibers, and even and odd sizes have different orbit signatures. The all-\(N\) coordinates in the layered-cube law therefore extend the three-layer state description rather than replace it. Their parity and orientation equations are stated for a labelled fixed-frame model; they are not obtained by treating every center sticker as one of the twelve ordinary edges.

                                                              The transition is consequently a specialization in one direction: the \(N=3\) outer-layer case of the family of layered cubes gives the ordinary fixed-frame action, while the abstract facet action explains why that outer turn moves precisely the selected surface fibers. Whole-cube rotations, marked centers, central slices and colour-only identifications are additional choices, not silent changes of coordinates.

                                                              What the identification supports

                                                                The explicit identification supports this chain of state descriptions:

                                                                1. incidence, flags and a local facet action explain what a move is;
                                                                2. the layered model and its first law explain which fibres and constraints appear as \(N\) changes;
                                                                3. the fixed-frame three-layer coordinates make the three familiar constraints and their constructive sufficiency explicit;
                                                                4. human methods and computer searches use the same state action while assigning different meanings to preservation, cost and optimality;
                                                                5. a declared move word can be replayed and checked against these state descriptions.

                                                                Three linked comparisons use the same distinctions. Label transport and branched unfolding study return maps on simplex labels; compatible quotients test whether an identification of sticker observations inherits a move; and scrambling and spectra study which functions retain memory under a declared move law. These are comparisons of transport, forgetting and randomization, not identifications with the fixed-frame cube action.

                                                                How the notes are organized [frap-000C]

                                                                  The developed mathematics follows one practical question: what data must a move carry so that we can later compose moves and test reachability? Incidence, symmetry and moves answers it in order. First, polytopes, flags and facet symmetry describe the shape and show their role in triangle and tetrahedron examples. Next, facet moves turn a local facet rotation into an action on incidence positions, with tetrahedron and cube calculations that make the construction concrete. Only then do positions, orientations and global constraints separate a piece's destination from transport in its local frame, while transfer and the effect of dimension connect that transport with a group homomorphism and compute the relevant local abelianizations.

                                                                  The cube examples above connect the general mathematics with solving and play. The later accounts use those concrete states and moves as their starting point:

                                                                  The transition between the abstract facet action and the fixed-frame cube coordinates is recorded in how the cube models connect. Read it before the all-\(N\) law if the distinction between incidence positions, cubie fibres and observations is new.

                                                                  1. General Rubik puzzles: the incidence actions, framed orientations and transfer developed below.
                                                                  2. The layered-cube family: piece types, layer turns and conventions as \(N\) varies.
                                                                  3. The layered-cube law: all-\(N\) state coordinates and a constructive characterization of reachability.
                                                                  4. The complete three-layer cube: explicit outer generators, impossible states, the standard count and a worked cleanup.
                                                                  5. Human methods and computer search: recognizable cases, algebraic effects, usability, move costs and optimality. The complete layer-first method supplies one full human construction, and computer solving and efficiency supplies exact small search, admissible bounds and subgroup reduction.
                                                                  6. From move sequences to play: notation, scrambling, recognition, replay and complete solves, including the worked replay and projected lower bound.

                                                                  Further connections relate the developed orientation and transfer calculations to label transport and topology and scrambling and search functions. The quotient-compatible move account gives the criterion for an identification of observations to inherit a puzzle action. Together these results compare transport, randomization and information loss around the same concrete move.

                                                                  Five encounters with a cube [frap-001C]

                                                                    Put a solved three-layer cube on a table and turn its right face once. The move is familiar, but it raises concrete puzzles: which labelled pieces moved, why can a visually plausible arrangement be impossible, and how can another person undo the turn without watching the first hand? Begin with these experiences, then introduce only the distinctions needed to explain them.

                                                                    One turn is larger than the visible face

                                                                      The shaded strip is only the visible part of the layer: the same stickers cross the front, right, back, top, and bottom faces; viewed from above, the top-face part is the corresponding rightmost strip. A whole-cube rotation would change the viewing frame, whereas this face move keeps the six centres fixed and changes one layer. The layered-cube family makes this distinction for every \(N\); the complete three-layer cube spells it out for the six outer generators.

                                                                      A sticker needs a home and a frame

                                                                        A player sees colours; a proof must also track each sticker's cubie and transported local frame. How the cube models connect follows one sticker through incidence position, cubie fibre, and fixed-frame coordinate. A colour quotient forgets labels, while a larger cube adds positions, so the state space changes with the question.

                                                                        A plausible state can still be impossible

                                                                          Take a solved cube and twist one corner in place. Every sticker still has a cubie-sized home, and the three colours at that corner look locally consistent. But the corner orientations now have sum \(1\) modulo \(3\), whereas every face turn preserves sum \(0\). No sequence of legal turns can produce this state.

                                                                          The complete three-layer cube derives the matching edge and permutation restrictions from the six generators. The calculation is useful at the table: it tells a solver when a scrambled-looking object was assembled illegally, and it tells a proof which state space is worth searching.

                                                                          A human algorithm has to preserve these legal states while it borrows pieces from completed areas.

                                                                          A formula often disturbs pieces that were already placed. The disturbance is acceptable only when the setup is undone and the completed pieces return. A working slot, a trigger, and an inverse setup make that promise visible. The complete layer-first method repeats this pattern through the cross, the first layer, the middle layer, and the last layer. Its completeness depends on the preservation and termination checks, not on the formula's familiarity.

                                                                          Undoing a scramble checks the notation

                                                                            The worked replay and projected lower bound fixes the viewing convention, records the ten tokens, and applies their inverse. Sticker replay returns to solved. A corner-only projected search finds no solution of depth at most two, so the evidence is the bounded statement

                                                                            \[3\le d_{\rm HTM}\le 10.\]

                                                                            The inverse word is an upper-bound witness; the projected search is a lower-bound witness. The computer account explains why these checks do not prove that ten is shortest. A state, a word, and a replay log are enough to test the notation before asking for a general solver.

                                                                            Changing one ingredient changes the problem

                                                                              Once the move, its frame, its local repair, and its replay are explicit, three variations separate the effects of enlarging the cube, changing the shape, and forgetting labels.

                                                                              The layered-cube family develops the first variation; the incidence and facet construction in incidence, symmetry and moves develops the second; and which moves survive a quotient? gives the descent condition for the third. Framed orientations, label transport and topology, and scrambling, spectra and search then study the extra information each variation introduces or forgets. Their hypotheses stay attached to the concrete move that prompted the question.

                                                                              A face turn can therefore be drawn, named, replayed, and checked before it is generalized. The resulting state maps and invariants support later formal certification or measured comparisons, while the examples here make clear which claim each later programme would have to establish.

                                                                              The first law for the layered cube [frap-0004]

                                                                                The layered model needs a reachability test. A tuple of permutations and orientations can describe a dismantled cube without being obtainable by legal slice turns. We now choose the labelled, fixed-frame state space from the layered-cube model and state the first law for that space. The theorem is a statement about labelled cubies; an ordinary colour pattern is its observation after equal-looking labels have been forgotten.

                                                                                Coordinates for one state space

                                                                                  Let \(K_o=(N-3)/2\) for odd \(N\) and \(K_e=N/2-1\) for even \(N\). For every cube there are a corner permutation \(\sigma \in S_8\) and corner orientation vector \(x\in (\mathbb Z_3)^8\). For odd \(N\), there is also a single-edge permutation \(\tau _s\in S_{12}\) and flip vector \(z\in (\mathbb Z_2)^{12}\). Each coupled-edge orbit indexed by \(i\in I\) has a permutation \(\tau _i\in S_{24}\) and a labelled orientation vector \(y_i\in (\mathbb Z_2)^{24}\). Each center-corner orbit has a permutation \(\rho _i\in S_{24}\). Center-edge orbits are indexed by a finite set \(J\); write \(\rho _{ij}\) for the orbit whose two distance signatures are \(i,j\). In the fixed-center convention, \(I=\{1,\ldots ,K_o\}\) and \(J=I^2\) when \(N\) is odd, while \(I=\{1,\ldots ,K_e\}\) and \(J=\{(i,j)\in I^2:i\ne j\}\) when \(N\) is even. This indexing records which coupled-edge signatures border a center-edge orbit; it is the coordinate version of the rings in the preceding chapter.

                                                                                  The full labelled configuration space is the product of these factors, with the solved state represented by identity permutations and zero orientation vectors. It is deliberately larger than the move group. For example, a vector \(y_i\) records all sticker orientations allowed by a labelled reassembly, even though the legal slice action fixes the relevant coupled-edge orientation in its orbit. Whole-cube rotations are still outside the generator set, and the six face directions remain part of the frame.

                                                                                  The invariant law

                                                                                    For odd \(N\), a configuration is legal exactly when

                                                                                    \[\begin {aligned} \operatorname {sgn}(\sigma )&=\operatorname {sgn}(\tau _s)= \operatorname {sgn}(\rho _i) &&(i\in I),\\ \operatorname {sgn}(\rho _{ij})&=\operatorname {sgn}(\sigma ) \operatorname {sgn}(\tau _i)\operatorname {sgn}(\tau _j) &&((i,j)\in J),\\ \sum _r x_r&=0\pmod 3,\qquad \sum _r z_r=0\pmod 2,\\ y_i&=0 &&(i\in I). \end {aligned}\]

                                                                                    For even \(N\), the corresponding law is

                                                                                    \[\begin {aligned} \operatorname {sgn}(\sigma )&=\operatorname {sgn}(\rho _i) &&(i\in I),\\ \operatorname {sgn}(\rho _{ij})&=\operatorname {sgn}(\sigma ) \operatorname {sgn}(\tau _i)\operatorname {sgn}(\tau _j) &&((i,j)\in J),\\ \sum _r x_r&=0\pmod 3,\\ y_i&=0 &&(i\in I). \end {aligned}\]

                                                                                    The empty-index cases are part of the statement: \(N=2\) has only corners, \(N=3\) has the usual twelve single edges and fixed centers, and \(N=4\) has one coupled-edge and one center-corner orbit but no center-edge orbit. The \(N=5\) instance has one orbit of each coupled-edge, center-corner and center-edge type, giving the 98 visible cubies counted earlier.

                                                                                    Necessity follows by checking generators. An outer face turn is a 4-cycle on corners, single edges, center corners and each relevant center-edge orbit, and two 4-cycles on each affected coupled-edge orbit. An inner slice turn is the identity on corners and single edges, and is a 4-cycle on its coupled edges and on the matching center-edge orbits, with two 4-cycles on center corners. These cycle signs give the two parity rows. The corner character changes by two opposite units on a non-\(U\)/\(D\) face and not at all on a \(U\)/\(D\) face, so its sum is fixed modulo 3. An outer turn flips four single edges, so their sum is fixed modulo 2. The chosen coupled-edge sticker convention makes every legal \(y_i\) zero. The solved state satisfies the rows, hence every word in the generators does.

                                                                                    Bonzio--Loi--Peruzzi give the labelled Professor's-Cube version of the same tuple and its corrected coupled-edge condition [bonzio2018nxnxn, Theorem 5]. Salkinder states the odd and even families and checks these generator effects in [salkinder2021nxnxn, Theorems 8--9]. Their notation indexes center-edge orbits by the two adjacent coupled-edge signatures; our \(J\) makes that indexing explicit. The sources also use a shifted notation for a middle slice when constructing a single-edge 3-cycle. That auxiliary word is a composition of parallel slice turns in the fixed-center model; it does not add a central generator to the law above.

                                                                                    Why the conditions are sufficient

                                                                                      The converse uses localized effects rather than a counting guess. First use a face turn, or an internal slice turn, to correct the common parity signs. The sign equations ensure that after this step every permutation factor is even. Commutators and conjugates then supply the required local subgroups:

                                                                                      • a corner 3-cycle, for example \([[R:U],D]\), generates \(A_8\) on corners;
                                                                                      • the single-edge 3-cycle \([F,[R:S]]\) generates \(A_{12}\) in the odd case, with the middle-slice symbol interpreted by the parallel-slice composition just described;
                                                                                      • \([[R_j^{-1},D_i],F^{-1}]\) gives 3-cycles on a center-corner orbit when \(i=j\) and on a center-edge orbit when \(i\ne j\);
                                                                                      • the analogous coupled-edge commutator gives \(A_{24}\) on each coupled-edge orbit. In even size the source's explicit odd permutation on that orbit upgrades it to \(S_{24}\); in odd size the parity row keeps it at \(A_{24}\).

                                                                                      Conjugating these words moves each 3-cycle through its whole orbit. Thus the local subgroups clear the corner, single-edge, center-corner, center-edge and coupled-edge permutations in turn. Finally, commutators that twist two corners in opposite directions generate the kernel of the corner-sum map \((\mathbb Z_3)^8\to \mathbb Z_3\); the corresponding edge-flip commutators generate the kernel of the single-edge sum. The coupled-edge orientation vector is already zero by its law. The parity and orientation equations are exactly what is needed for each cleanup step, so reversing the resulting word reaches every tuple satisfying the displayed conditions. This is the constructive sufficiency argument of [salkinder2021nxnxn, Theorems 10--18]; it is a subgroup generation argument, not an assertion that one universal short algorithm solves every size.

                                                                                      There are two boundaries to keep visible. The theorem concerns labelled positions and a fixed frame; a colour-only observation quotients by equal labels and may identify several legal labelled states. Also, Bonzio's and Salkinder's orbit labels are source conventions, so a numerical state count must be transported through the index sets before comparison. The next three-layer chapter specializes the same action to \(N=3\), where all center and coupled-edge factors disappear and the three familiar constraints can be worked through with explicit face generators.

                                                                                      Related

                                                                                        How the cube models connect [frap-001A]

                                                                                          An abstract cube turn records incidence positions and restricts a shape symmetry to one facet. A layered-cube turn records cubies, stickers, frames and allowed layers. For the ordinary three-layer cube these are two descriptions of the same action; the map below identifies their coordinates.

                                                                                          From facet incidence to the 54-sticker state

                                                                                            The facet calculation in one ordinary cube face turn selects the facet-colored part of the incidence set. Put the cube's surface cells at \(p\in \{-1,0,1\}^3\ \text {with}\ \lVert p\rVert _\infty =1\). A visible sticker is a pair \((p,n)\), where \(n=\operatorname {sgn}(p_i)e_i\) for each nonzero coordinate \(p_i\). Thus a corner has three visible normals, an edge has two, and a face center has one:

                                                                                            \[8\cdot 3+12\cdot 2+6=54.\]

                                                                                            The vertex--facet incidences in the facet-colored set map to the pairs above by sending a cube vertex, edge, or facet to its surface-cell coordinate and its incident facet normal. Conversely, the zero-coordinate pattern of \(p\) determines the cubie (three nonzero entries for a corner, two for an edge, one for a centre), and \(n\) determines the incident facet, so this is a bijection between the 54 incidences and the 54 pairs. The cubie counts and this incidence decomposition follow the general model in [bonzio2018nxnxn, Sections 2--3]; the displayed pairing and 54-count are the calculation used here. For \(N=3\), group these pairs into eight corner and twelve edge sticker fibres, leaving the six center positions as fixed frame marks. A legal outer-face move preserves this partition and carries the labels in a corner fibre or edge fibre to the corresponding destination fibre. Write \(C_p=\{(p,\operatorname {sgn}(p_i)e_i):p_i\ne 0\}\) for a corner fibre and \(E_p\) for an edge fibre. For a corner, order its normals as \(L_p=(u_p,f_p,r_p)\), where \(u_p=\operatorname {sgn}(p_3)e_3\), \(f_p=\operatorname {sgn}(p_2)e_2\), and \(r_p=\operatorname {sgn}(p_1)e_1\). For an edge, put first the U/D normal when \(p_3\ne 0\), otherwise the F/B normal, and put the other normal second. A move sends \(C_p\) or \(E_p\) to a destination fibre. Let \(k_p\in \{0,1,2\}\) be the index of the transported U/D normal in \(L_{p'}\). If the source cubie has \(p_3=+1\), set \(x_p=k_p\); if it has \(p_3=-1\), set \(x_p=-k_p\) in \(\mathbb Z_3\). Thus each side turn gives two \(+1\) and two \(-1\) corner twists, while U/D turns give zero, matching the U/D-sticker convention in the complete three-layer cube. For an edge, its distinguished normal agrees or disagrees with the first destination label, giving \(z\in \mathbb Z_2\). This explicit extraction fixes each per-fibre bijection and the destination permutation without asserting an unstated cyclic convention. Therefore it induces exactly the fixed-frame coordinates

                                                                                            \[(\sigma ,x;\tau ,z)\in S_8\times (\mathbb Z_3)^8 \times S_{12}\times (\mathbb Z_2)^{12}\]

                                                                                            used in the complete three-layer cube. More explicitly, if a move sends a corner fibre \(C_p\) to \(C_{p'}\), set \(\sigma (p)=p'\); the destination edge fibre similarly gives \(\tau \). For the positive-\(z\) turn \(q(x,y,z)=(y,-x,z)\) from one ordinary cube face turn,

                                                                                            \[((1,1,1),e_z)\longmapsto ((1,-1,1),e_z),\qquad ((1,1,1),e_y)\longmapsto ((1,-1,1),e_x).\]

                                                                                            This exhibits both the destination corner and the transported local frame. The induced map is a change of state description for the legal action: it sends a sticker permutation to its corner and edge destination permutations together with the local frame changes. It does not assert that every permutation of the 54 positions is a legal cube move.

                                                                                            This also separates two kinds of forgetting. Passing from the incidence action to the displayed coordinates forgets the individual sticker addresses inside each recorded local frame. Passing from labelled cubies to an ordinary colour observation forgets equal-looking labels. The latter is the observation quotient discussed in which moves survive a quotient, and it is a separate identification of states rather than a change in these coordinates.

                                                                                            What changes when layers are added

                                                                                              For larger \(N\), the same surface-cell description retains the corner and edge fibers but adds coupled-edge and center orbits. Inner layers move those new fibers, and even and odd sizes have different orbit signatures. The all-\(N\) coordinates in the layered-cube law therefore extend the three-layer state description rather than replace it. Their parity and orientation equations are stated for a labelled fixed-frame model; they are not obtained by treating every center sticker as one of the twelve ordinary edges.

                                                                                              The transition is consequently a specialization in one direction: the \(N=3\) outer-layer case of the family of layered cubes gives the ordinary fixed-frame action, while the abstract facet action explains why that outer turn moves precisely the selected surface fibers. Whole-cube rotations, marked centers, central slices and colour-only identifications are additional choices, not silent changes of coordinates.

                                                                                              What the identification supports

                                                                                                The explicit identification supports this chain of state descriptions:

                                                                                                1. incidence, flags and a local facet action explain what a move is;
                                                                                                2. the layered model and its first law explain which fibres and constraints appear as \(N\) changes;
                                                                                                3. the fixed-frame three-layer coordinates make the three familiar constraints and their constructive sufficiency explicit;
                                                                                                4. human methods and computer searches use the same state action while assigning different meanings to preservation, cost and optimality;
                                                                                                5. a declared move word can be replayed and checked against these state descriptions.

                                                                                                Three linked comparisons use the same distinctions. Label transport and branched unfolding study return maps on simplex labels; compatible quotients test whether an identification of sticker observations inherits a move; and scrambling and spectra study which functions retain memory under a declared move law. These are comparisons of transport, forgetting and randomization, not identifications with the fixed-frame cube action.

                                                                                                Example. One ordinary cube face turn [frap-000P]

                                                                                                  Realize the cube as \([-1,1]^3\) and let \(H\) be the facet \(z=1\). The facet move construction of [marchetti2025rubik, Definitions 5--6] gives the following explicit outer-face action. The shape rotation

                                                                                                  \[q(x,y,z)=(y,-x,z)\]

                                                                                                  is a clockwise quarter-turn when viewed from positive \(z\), with \(x\) rightward and \(y\) upward. Its restriction to \(H\) supplies our move. The corner \((1,1,1)\) goes to \((1,-1,1)\), and its color on \(y=1\) goes to the color position on \(x=1\). The corresponding corner at \(z=-1\) is fixed by the puzzle move, although the global map \(q\) moves it.

                                                                                                  The signs give the \(x,y\) coordinates of the four corners on \(z=1\). The arrows show their clockwise motion; the central incidence record \((H,H)\) stays fixed. No orientation of a marked center is recorded.

                                                                                                  Restricting the second entry to facets gives \(8\cdot 3+12\cdot 2+6=54\) incidence positions: three per corner, two per edge and one per face center. The turn moves twelve corner stickers and eight edge stickers, fixing the other thirty-four positions, including the unmarked center of the turned face. This is the usual outer-face action on visible stickers of a \(3\times 3\times 3\) cube in a fixed frame.

                                                                                                  The full incidence set has \(8\cdot 7+12\cdot 3+6=98\) elements: above each vertex are that vertex, three edges and three facets; above each edge are itself and two facets. The facet-colored subset is invariant under every move. Its restricted action supplies the comparison with ordinary stickers; a faithful restriction for all abstract polytopes is not needed. Marked center orientations and inner-layer turns are additional data, absent from this comparison.