King Otto’s birthday cake was a giant mousse. Its tiers were two concentric regular octagons. The upper tier’s span equalled the lower tier’s side-length (the diagrams show top and side views). Each tier’s height equalled the base tier octagon’s side-length (a two-figure whole number of inches under 20) divided by the square root of two (chef used a 45 degree right triangle gauge for this). When all tables at the party (arranged in three equal lines) were fully occupied (each set for the same number of couples) each person got an identical portion of eight cubic inches of cake (not necessarily in the shape of a cube). The small amount left over was less than this.

How many tables were there?