Draw a table with four rows and six columns. Make each of the twenty-four table fields square. Now draw the longest line along the edges of the fields such that it returns to its starting point without ever touching or crossing itself. There are many such patterns, which look like closed meanders or labyrinths, but they all form a circle, topologically speaking. The perimeter of the largest circle that can be inscribed in the table separates that which is inside from that which is outside. If you draw a number of such patterns, you will notice several interesting things. The perimeter of each circle is thirty-four field edges in length. Each circle encompasses sixteen table fields, leaving eight of them outside. The connections between the fields within the circle form a tree, topologically speaking again. A tree is a pattern without cycles or holes. The fields outside of the circle form trees, as well, and there will be between one and six of them in each pattern, the smallest of which will cover one field only. Now consider all the circles with longest perimeters that can be drawn on the table with four rows and six columns, including their horizontal and vertical mirror images, all of which are different because no pattern is symmetrical with respect to its horizontal or vertical axes. How many patterns are there? And how many such patterns are there for tables with any number of rows and columns?

Addendum (June 18, 2003)

The largest circle in a rectangle? Sounds familiar? Could it be a cross-section through the largest sphere in a box? Topologically speaking, again. The brain-and-skull problem, of course!