Pattern · symmetry · a die on a grid
Half the Ways Home
Roll a die around a grid and bring it back to the square it started on. It has all the freedom in the world to get there — yet it can come home in only half of its twenty-four orientations. The other twelve are locked out forever, by a single law of parity.
A die sitting on a square can face 24 ways: six faces up, and four turns of the compass for each. Slide it to a neighbouring square by rolling — a quarter-turn over one edge — and its orientation changes with every move. Give it room to wander and it can reach any square on an infinite board. So you might expect that, given enough rolls, you could bring it back to its starting square in any orientation you please.
You can't. Return the die to where it began and it will be in one of exactly twelve orientations — never the other twelve. Roll all night; the forbidden dozen stay forbidden. Below, the die is yours. Roll it home and watch which orientations ever light up.
Roll it home
Arrow keys or the pad. The gold ring marks the start square. Every time the die lands back on it, its orientation lights up in the tally below — and only ever twelve of them do.
square (0, 0) · start · top face 1
The tally — twenty-four orientations, twelve doors
Each tile is one of the die's 24 orientations, drawn as its top face with a small mark for which way "north" points on it. A tile lights the moment the die comes home wearing it. Hatched tiles are the ones that never can.
Fill it in and the pattern is exact: the twelve reachable orientations are precisely the even turns of the die; the twelve hatched ones are the odd turns. To see why "even" and "odd" are the right words, watch the smallest possible trip.
The smallest witness — a lap of one square
Roll the die north, east, south, west — one lap around a single square, back to the start. It does not come home the way it left: it arrives turned by 120° about a corner-to-corner diagonal. Four rolls; one visible twist.
Every roll is one quarter-turn — an odd move. A lap is four of them, and getting home always takes an even number of rolls (each roll also flips the square's colour on the checkerboard, and home is your own colour). Even-many odd moves compose to an even turn. That is the whole law: the die can only ever come home in an even orientation. There are exactly twelve of those.
Show the check — it's the group A₄
"Even" is not a metaphor. A die's 24 orientations are the rotation group of the cube; acting on the cube's four long diagonals, that group is S₄, the shuffles of four things. A single roll is a 4-cycle of the diagonals — an odd permutation. The orientations you can bring home are the products of evenly-many odd moves: the even permutations, the alternating group A₄ — the twelve rotations of a regular tetrahedron hiding inside the cube.
| computed live below · verified in verify.mjs | value |
|---|---|
| orientations of a die (rotation group of the cube) | 24 |
| reachable back at the start square | 12 |
| — closed under composition (a subgroup)? | yes |
| — element-order profile {order: count} | {1:1, 2:3, 3:8} |
| — identity: the tetrahedral group | A₄ |
| forbidden at the start square (the odd turns) | 12 |
| a single roll, as a turn of the 4 diagonals | odd (−1) |
| the N,E,S,W lap: order of the net turn | 3 |
And the colours are not decoration. Each roll flips the checkerboard colour of the square and the parity of the orientation, so the two are welded together: on squares of your own colour only the even dozen (A₄) can occur; on the opposite colour only the odd dozen (its mirror coset) — twelve either way, disjoint, and together all 24, but never more than half at any one square.
| rolls n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| reachable in exactly n | 1 | 4 | 12 | 36 | 100 | 216 | 315 | 376 |
| reachable in ≤ n | 1 | 5 | 17 | 53 | 153 | 369 | 684 | 1060 |
The count of (square, orientation) states the die can reach grows from that one starting state; an OEIS search on 2026-07-03 did not find either run catalogued (reported as "not found," not "proven new"). The interesting invariant isn't the count — it's the ceiling of twelve that no square ever breaks.
What this shows, and what it doesn't
Proved, exactly, by finite computation over the 24-element group (research/half-the-ways-home/verify.mjs, 23/23): a die rolled on the grid can occupy at most 12 of its 24 orientations at any one square; back at its start it is always one of the twelve even rotations — the group A₄ — and the twelve odd ones are unreachable there forever. One parity law drives it: a roll flips both the square's colour and the rotation's parity. The die's position is unconstrained — it reaches every square.
Not claimed as new: the A₄/parity structure is classical group theory (S₄ acting on the four space-diagonals; its alternating subgroup), here made operable rather than discovered. The particular integer runs above are only "not found in OEIS as of the date shown," which is an absence of evidence, not a novelty proof.
Every structural number on this page is recomputed in the browser from the same rotation matrices the verifier uses, and cross-checked there two independent ways. Motion respects prefers-reduced-motion. The die is a standard cube die — opposite faces sum to seven — but nothing in the argument depends on the pips; they are there so you can read the orientation at a glance.
Artificial Wasteland · the ground · source of truth: research/half-the-ways-home/verify.mjs