Puzzle No Tipping
This first puzzle comes from the field of
physics, Newtonian mechanics in fact.
As Archimedes famously observed and
as every child on a seesaw reconfirms,
if you put a heavy object far out on a
lever arm, it will exert a twisting influence around any support. That twisting influence is called “torque” and is
equal to the weight times the distance
(the angle also comes in, but that does
not concern us here). If the object is
to the left of the fulcrum, the direction of the torque is counterclockwise,
and vice versa. To compute the torque
around a support simply sum all the
torques of the individual objects.
For example, if a 10 meter board
weighs 2 kilograms and its center of
mass is at its middle, and we put a fulcrum at 3 meters, then the board will
twist clockwise with a torque of 3x2= 6
kilogram-meters. If we put a 5-kilo-
gram weight at the very left end of the
board, it will cause a counterclockwise
torque of 5x3= 15 kilogram-meters.
The net torque will be counterclockwise 15–6= 9 kilogram-meters.
Now to the puzzle: You are present-
ed with a straight, evenly weighted
board 20 meters long and weighing 3
kilograms. The middle of the board
( 10 meters from the left end) is the
center of mass. We call that position 0.
So possible positions range from – 10
to + 10. The board is supported at – 1. 5
and + 1. 5 by two equal supports both
10 meters from a flat floor.
Package Position
Weight
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
- 8
- 6
- 6
- 4
- 3
- 3
- 2
1
2
2
3
5
5
8
8
4
8
1
5
10
2
2
10
9
5
3
9
1
5
10
Find an order of removing packages
such that the board never tips. Tyler
Neylon has developed a two-person No
Tipping game which you can find at:
http://cs.nyu.edu/courses/fall05/
G22.2965-001/notipping.d/ index.html
—Dennis E. Shasha
Find the solution at: http://xrds.acm.
org/bemusement-solutions/ 2010.cfm
A Mathematical Paradox
PhD Comics ©Jorge Cham
Can you can do better? Bemusements would like your puzzles and mathematical games (but not Sudoku).
Contact xrds@acm.org to submit yours!
