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A Three Way Scale (Posted on 2005-09-05) Difficulty: 3 of 5
In the tradition of weighing puzzles, you have yet another batch of coins to sort. The only way to distinguish the coins is by their weights. This time there are nine coins. Three coins weigh 20g, three coins weigh 21g, and three coins weigh 22g.

You need to sort the coins by weight, but this time you must use an unusual three way scale. The scale operates as follows:

- Three weights must be placed on the scale for a meaningful result.
- If all three weights are equal, the scale will be in perfect balance.
- If one weight is lighter than the other two, the scale will show that weight is the lightest.
- If two weights are equal and are lighter than the third, the scale will indicate that scenario.

Ex1: If A=30 B=30 C=30 then the scale will be in balance.
Ex2: If A=30 B=31 C=32 then the scale will indicate that A is the lightest of the three, but not tell which of B and C is lighter.
Ex3: If A=30 B=31 C=31 then the scale will indicate that A is the lightest of the three, but not tell that B and C are equal. (undistinguishable from Ex2)
Ex4: If A=30 B=30 C=31 then the scale will indicate that A and B are equal and less than C.

Using this three way scale, can you sort the coins in five weighings?

See The Solution Submitted by Brian Smith    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Question Is 4 weighings possible? | Comment 2 of 3 |

There are 9!/(3!)^3 = 1680 possibilities for the identities of the 20- 21- and 22-gram coins.  Each weighing has 7 possible outcomes: In balance, A lightest, B lightest, C lightest, A&B lightest, B&C lightest, and A&C lightest.

The base-7 log of 1680 is 3.816.... Put another way, 7^4 = 2400 -- more than enough to cover the 1680 possible sets of identities, if the logistics can be worked out. Can a scheme be worked out so that enough combinations of four outcomes cover all possible sets of identities?  ... or is this impossible due to limitations on what can be put on the trays?

  Posted by Charlie on 2005-09-06 13:48:19
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