A multipan balance scale depicted below has one pan on the left and five on the right.

-+-       -+-  -+-  -+-  -+-  -+-
 |         |    |    |    |    |
 +----+----+----+----+----+----+
      |
      |

The pan on the left is a unit distance from the fulcrum. The pans on the right are 1, 2, 3, 4, and 5 units from the fulcrum. The balance scale is operated by placing an unknown weight on the left and placing some or all of a known set of weights on the pans on the right.

If the known weights are 1, 2, 3, 4 and 5, show that any integral weight 1 to 75 can be measured. What is the smallest measurement requiring 3 weights on the right? 4 weights? 5 weights?

If the restriction that only one weight can occupy a pan is added, show that any integral weight 1 to 55 can be measured. What is the smallest measurement requiring 3 weights on the right? 4 weights? 5 weights?

Proof by mathematical induction is the way to go!

Posted by elementofsurprize on 2009-01-20 20:31:53