# shift

Let*i*and

*j*be integers between -7 and 7. The

*basic shift transform*

**U**

^{i}

**V**

^{j}shifts a square

*i*squares to the right and

*j*squares up. (

**U**and

**V**are taken from the definitions of shiftvertical and shifthorizontal). Negative arguments shift in the opposite direction.

because ifU^{2}V^{3}[a1,d3] ≡ [c4,f6]

`a1`

is shifted right two squares and up 3 squares it becomes `c4`

; if `d3`

is shifted right two squares and up 3 squares, it becomes `f6`

.
As a special rule,
suppose a square designator contains an entire rank, like `[a-h2,c4]`

contains the entire second rank in addition to the `c4`

square. Then any right or left shifts of squares on that rank are ignored. Thus,

U^{2}V^{3}[a-h2,c4] ≡ [a-h5,e7]

That is, **U** has no effect on full ranks, and likewise **V** has no effect on full files.
A basic shift transform can be extended to a basic shift transform of an arbitrary filter: each square designator appearing in the filter is shifted by the specified amount.

Sometimes a basic shift transform will result in an empty square designator. For example,

U^{7}V^{7}[a-h2,c4]

will be empty. Any empty square designator makes the whole filter containing that empty square designator empty as well.

`shift`

is a transform filter that represents each of the 225 basic shift transforms including the identity.

The effect of `shift `

is first to create the 225 distinct filters that result from applying each of the basic shift transforms to *F**F*. Empty filters and repetitions are deleted, and the resulting set is called the *orbit* of `shift `

.
*F*

For example, the orbit of `shift ♔a1`

is 64 distinct filters, `♔a1`

, `♔b1`

, and so on.

On the other hand, the orbit of `shift {♔a1 ♚a8}`

only contains 8 filters, namely `{♔a1 ♚a8}`

,
`{♔b1 ♚b8}`

and so on. Any vertical shift results in one of the kings going off the board, and which makes the corresponding filter empty.

If *F* is a numeric filter then so is `shift `

in which case its value is the maximum of the values of the matching transformed versions of *F**F*.

shift countF

is a numeric filter whose value is the number of distinct matching transformed versions of *F*. For example, if the current position is the initial position, then

`shift count find ♕a1`

is the number of different squares on which there is a white queen in the game.