The Centrifuge
A centrifuge has 12 locations for test tubes, located evenly in a
circle (let them be referenced as the hour locations on a clock.)
Prior to spinning the tubes, the centrifuge must be evenly
balanced about its center. Assuming all test tubes weigh the same
amount, what numbers of test tubes (from 1 to 12) can be spun without
needing to add dummy tubes to balance out the weight? For example,
you can't spin 1 tube alone, but can you spin 5 tubes?
Source: rec.puzzles. Used as Ken's Puzzle of the Day 6/6/94.
Solutions were received from
Larry Baum,
Robert T. McQuaid,
Carlos Rivera.
Except for 1 and 11, any number of tubes from 1 to 12 can be balanced.
Any even number of tubes can be spun simply by placing pairs of tubes in
opposite locations on the centrifuge.
For 3 tubes, place the tubes in the
12, 4, and 8 positions (evenly spaced.)
For 9 tubes leave those positions empty instead.
For 5 tubes, add two tubes at opposite
locations to the solution for 3 tubes (at the 3 and 9 positions.)
For 7 tubes, leave all of these locations empty instead.
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