PRIZE PUZZLE FOR DECEMBER 2006
SIX POINTS ON A MOBIUS STRIP
The puzzle this month invites you to draw noncrossing 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.
Please send your answer to me, David Broughton, by email to
davidb67@clara.co.uk; closing date: 3rd January 2007.
Click this symbol
for my email address.
