By now you can probably see what's going on. We'll position the first three
blocks, as a unit, atop the fourth block exactly as they were positioned atop
the table. And we'll extend the fourth block one-eighth of the way off of the
table. The center of gravity of the fourth block is given by:
| cg4 := (3*(cg3 + 1/8) - 3/8) / 4; |
Once again, it is exactly zero. We can continue in this fashion indefinitely.
The top block extends 1/2 unit beyond the second block, which extends 1/4 unit
beyond the third block, which extends 1/6 unit beyond the fourth block, which
extends 1/8 unit beyond the fifth block, and so on down to the table. Thus, if
we have, say, six blocks, their total extension will be
Maple will easily compute the exact total extension:
| 1/2 + 1/4 + 1/6 + 1/8 + 1/10 + 1/12; |
If we have n blocks, of course our total extension will be
or, more precisely,
The question remains: as the number of blocks gets large, what happens to this
sum?
Joseph L. Zachary
Hamlet Project
Department of Computer Science
University of Utah