Here I'll report some findings of François Glineur on several questions about figures of constant width.

François Glineur formulated the problem of finding a figure of constant width as a linear programming feasibility problem with binary variables. Then he used a commercial software to get the solutions. For chessboards of size up to 20 the solver was capable in a few seconds of finding a solution or deciding that none exists. Here's a summary of the results.

1. Figures of constant width of type (n, n, w) with 1 <= n <= 20 exist for:

   w=1	n=1,4,...,20
   w=2	n=4,6,8,...,20
   w=3	n=11,...,20
   w=4	n=12,14,...,20
   w=5	n=17,...,20
   w=6	n=20
   w=7	none

2. There is only one figure of type (12,12,4). Here it is:

   

3. There are no figures of type (23,23,7). The solver was not able to decide if there is a figure of type (24,24,7).

The figure of type (20,20,6) answers a question made in my note "Stuff left out of the paper". The fact that there are no figures of type (23,23,7) invalidates the conjecture made in that same note saying that there is a figure of type (3w+2,3w+2,w) for all w. The figure of type (12,12,4) disproves the conjecture that there is no figure of type (n,n,w) for n<3w+2.

Imposing the restriction that the figures have central symmetry the solver was able to give the following results.

1. The are no figures of type (n,n,7) with central symmetry for n<=24. There is a figure of type (25,25,7) with central symmetry.

2. The are no figures of type (n,n,8) with central symmetry for n<=27. There is a figure of type (28,28,8) with central symmetry.