PRIZE PUZZLE FOR DECEMBER 2006
SIX POINTS ON A MOBIUS STRIP
The puzzle this month invites you to draw non-crossing lines between a number of points so that all points connect with all other points in the diagram. For example, if we have four points we can draw connecting lines that do not cross one another (and do not go "through" points) as shown in FIG 1. It is not possible to have more than four points in a simple plane.
However, if the points are on a Mobius Strip it is possible to connect up to six points. That is the puzzle. FIG 2 shows six points on a Mobius strip. A Mobius strip is a simple loop containing one twist as illustrated in FIG 3. The twist is not shown in FIG 2 because we are constrained to draw in two dimensions; but the two lines with arrows show where the strip joins after looping round. The arrows show how the lines representing the ends join up.
So the puzzle is to draw connecting lines between the six points of FIG 2 that do not cross one another and where each point is connected through the connecting lines to every other point. Obviously, one or more connecting lines will have to pass round the ends. You must label each line with some number or letter notation and, of course, the sequence (if there is more than one) of connecting lines at one end must match in reverse direction the sequence at the other end of the strip. So lines identified as, say, A,B,C at one end should match the sequence C,B,A at the other end (assuming three such lines, which is not necessarily the number you need).
Note that you must regard the points and lines as being within the surface, accessible from both sides.