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A Leaning Tower
© Copyright 1999, Jim Loy
Here we see five identical CD cases (CD
cases work a little better than dominos) balancing on the edge of a table. We
are seeing the short edge of each case, to ensure that the center of gravity of
each case is at its center. The top CD case is hovering over thin air. At first
thought, most people think that this is impossible, that we cannot stack the CD
cases like this. But, it is fairly easy. As I drew this, I put the center of
gravity of the top case directly over the edge of the second CD. In reality,
you may have to back off from this slightly, so that a speck of dust won't
upset the balance. So, you may need five or more CD cases to accomplish this
seemingly impossible balancing act.
It takes a little care to do this trick. Move the top CD case as far to
the right (in the diagram) as you can, over the second case. Then move those
two cases as far to the right as you can. Then do the same with the top three
cases, then the top four cases, etc. Finally move the bottom case over the edge
of the table, as far as you can. I have shown the theoretical fractions of a CD
case which you will be moving each case, in the diagram.
The wrong fractions are given in the excellent book 50 Nifty Science
Fair Projects, by Carol Amato and Eric Ladizinsky. They give 1/2, 1/4, 1/8,
1/16, etc. It takes a little simple physics (or arithmetic) to find the
position of center of mass of several of the CD cases. I won't bother with that
here.
The fractions in my diagram relate to the famous
Harmonic Series: 1/1+1/2+1/3+1/4+... Our
fractions are just half of these terms, and so is the sum. The amazing thing
about this series is that if you take it out to an infinite number of terms,
the sum is infinite. Some series do not sum to infinity. The series,
1/2+1/4+1/8+1/16... sums to one, for example. So, if we use enough CD cases,
finitely many, we can make the top case be any distance we want past the edge
of the table. And it will still balance.
Addendum:
If the CD cases lean out over the edge of a table,
then it only takes four CD cases to make one of them hover over thin air, the
table taking the place of my bottom CD case. If we allow a CD case to be
rotated somehow, then we get this simple solution, in which two CD cases are
sufficient, a trivial solution. I forgot to disallow this kind of solution.
Let's disallow it.
The excellent book, Mathematical Puzzles for the Connoisseur
(which contains some puzzles where you may have to look up the definitions of
pounds, shillings, and pence), by P. M. H. Kendall and G. M. Thomas has a
strange mistake. It asks, "Can you place one cube upon another, upon another,
etc., until the plan view [the view from above] of one of them lies entirely
outside the area of the base cube?" The obvious answer is "yes," and we can do
it with five cubes (not optimal), just like our CD boxes. The nonoptimal
solution given in the book is that it requires at least five cubes, as in the
first picture above right. From that we see that rotating a cube 45 degrees is
allowed here. And we can improve the book's solution to four cubes, as shown in
the second picture above right. It is the same solution as our CD cases, but by
rotating the top cube, we have improved the solution.
Another error in the same book (Mathematical Puzzles for the
Connoisseur) is the following puzzle: "What is the minimum number of
pennies that can be placed upon a table so that each penny touches three, and
only three, others? All the pennies must lie flat on the table." The book gives
the incorrect answer, 20. To the left is my solution with 16 pennies. To the
right is my solution, using dollars (my idea of a joke).
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