THE POTATO PUZZLE.
(
Various Dissection Puzzles)
Take a circular slice of potato, place it on the table, and see into how
large a number of pieces you can divide it with six cuts of a knife. Of
course you must not readjust the pieces or pile them after a cut. What
is the greatest number of pieces you can make?
[Illustration:
--------
/ 1/
/ 2 / 3 /
/ / /
/ 4 / 5/ 6 /
| / / / |
7/ 8/ 9/10 /
/ / / /
/11/12/13/
/ / /
/14/15/
/ /
/16/
-----
]
The illustration shows how to make sixteen pieces. This can, of course,
be easily beaten.
Answer:
As many as twenty-two pieces may be obtained by the six cuts. The
illustration shows a pretty symmetrical solution. The rule in such cases
is that every cut shall intersect every other cut and no two
intersections coincide; that is to say, every line passes through every
other line, but more than two lines do not cross at the same point
anywhere. There are other ways of making the cuts, but this rule must
always be observed if we are to get the full number of pieces.
The general formula is that with n cuts we can always produce (n(n +
1) + 1)/2 . One of the problems proposed by the late Sam Loyd was to
produce the maximum number of pieces by n straight cuts through a
solid cheese. Of course, again, the pieces cut off may not be moved or
piled. Here we have to deal with the intersection of planes (instead
of lines), and the general formula is that with n cuts we may produce
((n - 1)n(n + 1))/6 + n + 1 pieces. It is extremely difficult to "see"
the direction and effects of the successive cuts for more than a few
of the lowest values of n.
