10. The Touching Cigarettes

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FOUR GOLF BALLS can be placed so that each ball touches the other three. Five
half-dollars can be arranged so that each coin touches the other four.
Is it possible to place six cigarettes so that each touches the other five? The
cigarettes must not be bent or broken.

The answer

There are several different ways of placing the six cigarettes. The figure on
the left shows the traditional solution as it is given in several old puzzle books.
To my vast surprise, about fifteen readers discovered that seven cigarettes
could also be placed so that each touched all of the others! This of course makes
the older puzzle obsolete. The figure on the right, sent to me by George Rybicki
and John Reynolds, graduate students in physics at Harvard, shows how it is
done. “The diagram has been drawn,” they write, “for the critical case where the
ratio of length to diameter of the cigarettes is . Here the points of contact
occur right at the ends of the cigarettes. The solution obviously will work for any
length-to-diameter ratio greater than . Some observations on actual
‘regular’ size cigarettes give a ratio of about 8 to 1, which is, in fact, greater than
, so this is an acceptable solution.” Note that if the center cigarette,
pointing directly toward you in the diagram, is withdrawn, the remaining six
provide a neat symmetrical solution of the original problem.
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