**n-Dimensional Hypercube or n-cube (Q_{n})**
It is a graph containing

** Snake (also n-snake)**
It is an induced, achordal path in an

- every two nodes, or vertices, that are consecutive in the path must be adjacent to each other in the hypercube,
- every two nodes that are not consecutive in the path must not be adjacent to each other in the hypercube, and
- the beginning node is not adjacent to the ending node (it’s an open path).

** Coil (also n-coil)**
It is a closed path (or cycle) which resembles a

** Maximal Snake**
It is a

** Transition Sequence**
To make a transition sequence you must first name the bits in the node numbers. Typically the bit representing

- for non-cyclic paths there are two possible transition sequences: starting from either end write down the order in which the bits change from node to node.
- for cycles with
*m*nodes there are*2m*possible transition sequences: pick a start node and a direction and write down the order in which the bits change.

Each transition identifies the edge from one node to its neighbor based on the dimension change. Some authors prefer using transitions numbered *1* to *n*, while other authors prefer transitions numbered *0* to *n-1*. The former is slightly more intuitive (an *8*-dimension hypercube using transition *8* to move into the *8 ^{th}* dimension, while the latter corresponds to binary numbering). There is also some variation among publications related to numbering the transition from left to right or from right to left.

** Symmetric Coil**
It is identified as having a

** The State of the Hypercube**
A set of

`ON`

or `OFF`

. If all the `ON`

nodes can be touched exactly once by starting at one of them and making moves to adjacent nodes then the hypercube state is a `OFF`

nodes since they are the ones that aren't `ON`

, and note that even if we jumble up the order in which we list the `ON`

nodes, the path they make is still uniquely defined since we use a fixed adjacency function of Hamming distance
Some **operations** on the state of the hypercube preserve the **achordal path constraint**:

**Translation**- XOR each`ON`

node with a fixed bitmask to produce a new set of`ON`

nodes. There are*2*possible translations corresponding to shifting the starting node to each of the^{n}*2*nodes in the hypercube. Translation preserves the order in which the dimensions are used.^{n}

**Rotation**- Consistently reorder the bits in all the`ON`

nodes. There are*n!*possible rotations corresponding to the*n!*possible ways to order the*n*bits. Nodes*0*and*(2n)-1*are invariant under all*n!*rotations. In general, a node is invariant under*(w!)((n-w)!)*rotations where*w*is the number of bits set to*1*in its node number. (A node's degree of rotation invariance might be an interesting node metric to investigate).

** Equivalence Class**
The set of states you get by applying all

** The Form of a Transition Sequence**
Remove all but the first occurrence of each dimension from the transition sequence. In other words, the form is the order in which the snake “uses” the dimensions, where the “usage” order of a dimension is determined by the first transition in the snake for that dimension relative to the other dimensions. Note that there are

** Canonical Form**
A restriction on transition sequences such that dimension

- for non-cyclic paths:
*c = 2* - for cycles with
*m*nodes:*c = 2m*

Canonical snakes will always start {*0,1,2*} in transition sequence and {*0,1,3,7*} in node sequence. Two snakes with the same canonical form are said to be synonyms.

** Rewrite in Canonical Form**
Any transition sequence can be rewritten in canonical form as follows: create a map from its form to canonical form. Use the map to re-encode the transition sequence of the path so that it is in canonical form. For example, if a path's transition sequence starts with a

**s-Fold Symmetry**
If canonical form admits

** Palindrome**
A non-cyclic path having maximum (

** Reversal**
As an operation on the hypercube state, reversal is just one of the possible

`ON`

nodes as you started with). Reversal is not defined for cycles since there is no start or end.