Here is a puzzle that consists of three pieces and that you can assemble
in two ways: you can make a circle or you can make a circle plus one extra point.
You create it as follows:
take the circle x^{2}+y^{2}=1 in the plane and let *p*
be the point with coordinates (1,0).
Our first piece Q_{1} consists of just the point *p*.
The scond piece Q_{2} we make by rotating *p* through
1, 2, 3, ... radians, so
Q_{2}={( cos n, sin n) : n=1,2,3,4,...}.
The third piece Q_{3} is simply what is left of the circle.

Clearly Q_{1}, Q_{2} and Q_{3} can be put together to form
exactly the circle.
If we rotate Q_{2} clcokwise through 1 radian then we have covered
the circle exactly but using Q_{2} and Q_{3} only,
the piece Q_{1} remains.
This works because all points (cos n,sin n) are different
(in particular *p* does not belong to Q_{2});
because of this Q_{2} basically plays the role of P_{1} from
the previous puzzle.

The form of this puzzle makes clear that you will not find it in the stores:
no knife is sharp enough to separate Q_{2} from the rest of the circle.
Try and plot Q_{2} on a graphic calculator by having it plot the points
( cos 1, sin 1) ... ( cos 100, sin 100);
you will see that Q_{2} is everywhere dense on the circle.

We can improve this puzzle so that even Q_{2} can be missed.
We do this by splitting Q_{3} into two pieces Q_{3a} and
Q_{3b}.
First we make Q_{3a} by rotating Q_{2} through
sqrt2 , 2sqrt2 , 3sqrt2 , ...
radians and Q_{3b} is what remains of Q_{3}.
We have covered the circle with Q_{1}, Q_{2}, Q_{3a}
and Q_{3b} but we can also do this with Q_{1} , Q_{3b}
by rotating Q_{3a} clockwise through sqrt2 radians;
Q_{2} is then no longer needed.

We see that in some puzzles we can put some pieces aside and still complete it. With the circle this is the best you can expect; because the circle can be rotated in one direction only we will not be able to double it. The ball and the sphere are different; by mixing two directions you can create the puzzle pieces of Hausdorff, Banach and Tarski.

k.p.hart@its.tudelft.nl Last modified: Wednesday 26-02-2003 at 10:35:46 (CET)