A *cylindrical board* is a normal board in which you glue column
**a** and column **h** side by side (resp.
row **1** and row **8**) in order to obtain
a vertical (resp. horizontal) cylinder.

On a vertical cylinder, it is possible to pass e.g. from column
**h** to column **a**, so that a game could
start with 1. e2-e4 e7-e5 2. Qd1-f3 Bf8xh2 3. Qf3-f3! Here the last
Q move is a neutral move along the empty 3rd row.

This fairy board has been invented for standard "forward" problems,
but it can be used to obtain special effects in retros. Of course, horizontal
cylinders are (almost) never used in retros because the initial position
would have the two Kings in contact.

The two huge treatises by Ceriani (see
booklist) offer several retros on vertical
cylinders.

## Castling on a Vertical cylinder

There are two extra castling possibilities on a vertical cylinder: The
king can castle queenside with his kingside rook, and he can castle kingside
with his queenside rook!
For example, if a1, b1, c1, and d1 are empty, if Ke1 and Rh1 never
moved, and if c1, d1, and c1 aren't observed by a black piece, then
white can castle queenside by moving his king to c1, and the rook from
h1 to d1.

## Looking for the h-a border

On a vertical cylinder, a position can almost always be shifted one column
to the right and still offer exactly the same potential futures. Therefore,
once you have composed a tricky cylindrical mate in 2, you may shift
your position in order to make maximal use of the deceiving "border"
crossing effects.
But then, retrograde analysis can sometimes be used to prove that
the board has been shifted ! This suggests a special kind of retro stipulations:
given a position on a cylindrical board, find where lies the h-a border.

(This is another genre of reconstruction
problems. Formally, the board should have all white squares to prevent
solvers from assuming that only even shifts are to be considered.)

**Luigi Ceriani**

La Genesi delle Posizioni,
1961

Ded. D. E. Cohen & F. R. Oliver

12+12. Vertical cylinder. Where is the
h-a border? What was the 1st move of the black QB?

Here we write a' to h' to denote the columns *as they appear on
the diagram*.

W captured all four missing black men with a'2xb'3 and h'2xg'3xf'4xe'5.

The black K could only enter the south cage through d'3, when the
white pawns were on c'3, d'2 and e'3. Thus Bc1 really is a promoted
B while the original black B on black squares (KB **or**
QB) did leave his initial square, which really was f'8 or h'8, and got
captured by the h' white P.

Three of the four black captures are b'7xa'6, g'7xh'6 and f'3xe'2
to reach the promotion square e'1.

The only way to unlock the south cage is to take back **Rg'1-e'1+,
Ke'1xBe'2 and Bf'1-e'2+**, explaining the fourth capture by black.
Thus the white B on black squares has been captured by a black P (on
h'6), so that its home square could not have been c'1 or e'1 inside
the south cage. It was a'1 or g'1.

Assuming this home square was g'1, we get g'8 as the home square of
the black B on white squares, so that the home square of the other black
B is b'8 or d'8, from which it could not have participated in the required
captures. Thus the home square really was a'1.

We deduce that the black B's are from a'8 and f'8, and that the h-a
border really is the c'-d' line. The first move of the black QB was
Bf'8-g'7 (i.e. Bc8-d7) and not Bc'8-b'7 as it would appear at first
sight.