FWIW, Bonzo, squaring the circle was one of the three classical problems of ancient math, often known as quadrature. The others were trisecting an angle (trisection), and doubling the volume of a given cube (duplation). As you mention, late in the 19th century it was proven that all three problems are not solvable with classical tools  an unruled straightedge and a "collapsing" compass. (That is, you could use the compass only to make arcs and circles, not for transferring a line length to another place.)
So don't worry  even after your clever puzzle/toy, you needn't run out of challenges.
