Page:Amusements in mathematics.djvu/189

164.—THE POTATO PUZZLE.

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 ${\frac {n(n+1)}{2}}+1$ pieces. 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 ${\frac {(n-1)n(n+1)}{6}}+n+1$ 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 $n$.

165.—THE SEVEN PIGS.

The illustration shows the direction for placing the three fences so as to enclose every pig in a separate sty. The greatest number of spaces that can be enclosed with three straight lines in a square is seven, as shown in the last puzzle. Bearing this fact in mind, the puzzle must be solved by trial.