4. Generalizations.

4.1 Figures of constant width in many directions.

Instead of row, columns and diagonals, we now look for figures of constant width along a predetermined set of directions. Here I'll briefly sketch their construction.

A direction of the chessboard is determined by a vector d=(a,b) in x (we identify the infinite chessboard with x ). The lines of the chessboard along this direction are the sets

Definition. A figure of constant width w along the set of directions D is one such that every line with direction d in D intersects the figure in exactly w squares.

In order to build figures of constant width on a given set of directions I'll make use of the notions of composition of figures and of extended constant width . The reader is referred to [Her-Rob] and [Riv-Var-Zimm] for the definitions. The following proposition is easily verified.

Proposition 1. If A has constant width v along d and B has extended constant width w along d then A composed with B has constant width vw along d.

Let D be a given set of directions. For d in D let F(d) be the figure {kd, k=0,1,2,...,w-1}. Suppose that d is in the first quadrant of x. We can always enclose this figure inside a chessboard of size N(d)x N(d) (N(d) chosen big enough), so that F(d) has extended constant width w along the direction d and extended constant width 1 along all the other directions. We now take F to be the composition of all figures F(d), d in D. By Proposition 1 the figure F has constant width w along all directions in D. Here is an example with w=2 and D={(0,1),(1,2),(1,-1)}:

Remarks.
1. The operation of composition is not associative. Thus, in order that F be well defined we need to specify the order in which the compositions are taken. No matter what order we choose, the resulting figure will have of constant width.
2. Our construction of figures of constant width along the set of directions D is far from proving a theorem like Theorem 1. In particular, it is not clear whether there are figures of type (n,n,...,n,w). Unfortunately, the proof of Theorem 1 cannot be generalized.

4.2 Figures of constant width on a toroidal chessboard.

There is a simple arithmetical obstruction to having a solution of the n-queens problem on an nxn toroidal chessboard. Namely, n must be relatively prime to 6. In the same way, a figure of type (n,k,w) in an nXn toroidal chessboard...

4.3 Figures of constant width on an n-dimentional chessboard.

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