Generator

The permutation (1 3 4 2) drawn as 4 strings, crossing 3 times

A generator in the algebra library, called 42 times across 8 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

wiring is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

The permutation (1 3 4 2) drawn as 4 strings, crossing 3 times. A permutation drawn as strings running from a row of numbered pegs to another, with every place two strings cross marked, and the crossing count checked against the number of pairs that are out of order.

A 3×3 determinant as six signed products, totalling −43

One permutation routed two ways: 3 crossings and 5

(1 3 4 2) as 3 swaps and as 5

Every permutation of 4 places, by sign

A 2×3 sliding puzzle: 360 arrangements of 720 can be reached

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

exactly half the arrangements of the 3×3 tray are reached ×5
the homomorphisms to the cyclic group of order 2 number 2 ×5
the class of shape 3+1 has 8 members by the formula too ×4
the class of shape 2+1+1 has 6 members by the formula too ×3
all 24 permutations are listed ×2
all 24 permutations of 4 places are listed ×2
the class of shape 1+1+1+1 has 1 members by the formula too ×2
the class of shape 4 has 6 members by the formula too ×2
the search on θ, inner paths 2, 2, 2 agrees with Wilson's theorem ×2
a single closed tour moves the tiles in one cycle ×1
a tour of 4 squares cycles 3 tiles ×1
a tray of four to nine squares, so its far end can be drawn ×1
a tray of four to ten squares ×1
adding a detour changes the crossing count by an even number ×1
and at least one target admits the sign ×1
and exactly two of the squares are unit squares ×1
and exchanging two is an odd permutation ×1
and its sign is what a cycle of that length has ×1
and no face turn changes the joint parity of the two permutations ×1
and one of the two is the sign ×1
and the half they generate is the even permutations ×1
and the other is the odd side ×1
and the sign is constant on it, which is why the sign is a function of the shape ×1
and the squares of their dimensions add to the size of the group ×1
and they are the same size ×1
because a single exchange is odd ×1
between five and two hundred scrambles ×1
between twenty and two thousand scrambles ×1
each scramble is between five and two hundred turns ×1
each scramble is between ten and four hundred moves ×1
each step of the tour follows an edge ×1
each step of the tour moves the blank one square ×1
every board one legal move from a reachable one has the same parity ×1
every distance fits in a byte ×1
every one of the eighteen turns was tried at every position ×1
every reachable arrangement has the same sign-and-blank-distance parity ×1
exactly half of the 720 arrangements can be reached ×1
exactly half the permutations are even ×1
exactly two of them are one-dimensional ×1
exactly two tokens have moved ×1
exchanging two tiles puts the board on the other value of the invariant ×1
from every arrangement some move goes one step closer ×1
no face turn flips the edges as a whole ×1
no face turn twists the corners as a whole ×1
no legal move leaves the component it starts in ×1
one component is the even side ×1
one corner twisted breaks its own invariant ×1
one corner twisted breaks only its own invariant ×1
one edge flipped breaks its own invariant ×1
one edge flipped breaks only its own invariant ×1
swapping two places flips the sign of every permutation ×1
swapping two tiles leaves the puzzle unsolvable ×1
the bars are split by nothing or by the blank's square ×1
the blank comes back to where it started ×1
the board is between nine and sixteen squares ×1
the board is between two and twelve squares ×1
the board is small enough to exhaust — at most six squares ×1
the characters are orthonormal under the group's own average ×1
the class of shape 1+1+1+1+1 has 1 members by the formula too ×1
the classes account for every permutation exactly once ×1
the commutators generate exactly half the group ×1
the commutators themselves are inside the subgroup they generate ×1
the cyclic groups tested have orders between two and eight ×1
the detour is between 1 and 3 extra loops ×1
the empty vertex is home again ×1
the empty vertex is where the tour says ×1
the even classes together are exactly half the group ×1
the far end holds one to four arrangements ×1
the farthest arrangement chosen exists ×1
the graph is one this family knows ×1
the group is on three, four or five places ×1
the identity's column holds each representation's dimension ×1
the invariant is the same at both ends of the tour ×1
the long decomposition is the same permutation ×1
the matrix is 3×3 with whole entries no larger than 20 ×1
the number of moves has the parity of the blank's displacement ×1
the number of swaps has the parity of the crossing count ×1
the permutation sends each of its 4 places somewhere different ×1
the reachable side has twelve arrangements ×1
the route is between three and nine squares of the board ×1
the search on a house agrees with Wilson's theorem ×1
the search on a six-cycle agrees with Wilson's theorem ×1
the search on a six-cycle with one chord agrees with Wilson's theorem ×1
the search on a wheel of six agrees with Wilson's theorem ×1
the search on K₂,₃ agrees with Wilson's theorem ×1
the search on the 2×3 tray agrees with Wilson's theorem ×1
the search on θ₀, inner paths 1, 2, 2 agrees with Wilson's theorem ×1
the second decomposition is 1 to 3 swaps longer, in pairs ×1
the short decomposition really is this permutation ×1
the sign of the arrangement alone does change, so it is not by itself the invariant ×1
the signed sum over permutations is the determinant ×1
the solved board has parity zero ×1
the squares are drawn for three, four or five places ×1
the squares of the dimensions add to the size of the group ×1
the strings cross exactly as often as the pairs are out of order ×1
the table is between 3 and 8 columns wide ×1
the table is drawn for 2, 3 or 4 places ×1
the table is drawn for three, four or five places ×1
the taxicab sum and the distance have the same parity ×1
the taxicab sum never exceeds the true distance ×1
the tour starts and ends at the empty vertex's home ×1
the tour starts from the blank's home square ×1
the two components account for every arrangement ×1
the view is one the family draws ×1
the walk reaches solved in exactly 31 moves ×1
the whole graph is drawn only for the two-by-two board ×1
there are exactly as many irreducible characters as conjugacy classes ×1
three to ten graphs ×1
three to ten vertices, so the search fits in memory ×1
two decompositions of one permutation have the same parity ×1
two pieces exchanged breaks its own invariant ×1
two pieces exchanged breaks only its own invariant ×1
two to six trays ×1
with nothing odd among them ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Algebra

A shared root, found without finding it

Two polynomials have a root in common exactly when one determinant built from their coefficients is zero. No root is computed, nothing is approximated, and the same construction turns two equations in two unknowns into one equation in one.

Algebra

The crossings that will not come out even

Draw a rearrangement as strings from one row of pegs to another and count where they cross. The count depends on how the strings are drawn; whether it is odd or even does not, and that single bit is what makes determinants exist and a sliding puzzle unsolvable.

Algebra

The only bit that survives

A shuffle can be called even or odd, and the label behaves under composition. Ask whether some cleverer label — a number out of three, or out of four — could behave the same way, and the answer is that nothing else can — one bit is exactly what a permutation gives up.

Algebra

The puzzle that is exactly half solvable

A sliding puzzle sold with two tiles swapped is not a hard puzzle; it is an impossible one, and the proof is a quantity that no slide can change. The same argument, run three times at once, says that one arrangement of a scrambled cube in twelve is reachable.

Number

The symbol is the sign of a shuffle

Multiplying every residue modulo p by a fixed number rearranges them. That rearrangement is a permutation, permutations have a sign, and the sign is exactly the Legendre symbol — so a question about squares becomes a question about crossings.

Algebra

Thirty-one moves from solved

Parity settles which half of a sliding puzzle's arrangements can be reached and is silent about how far away any of them is. Searching every reachable arrangement of the three-by-three tray answers the second question exactly — two arrangements sit thirty-one moves out — and parity turns up again, this time as a law about distance.

Algebra

When the label may be a matrix

A permutation carries exactly one bit into any commutative target, and commutativity is the restriction doing all the work. Drop it — let the label be a matrix — and what survives is a short finite table, computed here from traces and checked for orthogonality over every pair of rows.

Algebra

Which graphs let the tokens go anywhere

A sliding puzzle is a graph with a token on every vertex but one. Richard Wilson found in 1974 what every such puzzle can reach, and the answer has a surprise in it — the half the tray is stuck with is not a fact about permutations at all, but about the board being two-coloured — and one exception, a graph of seven vertices that reaches exactly 120 of 720.

The whole library · What the figures prove