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.

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.

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.