perplexus.info :: Weights and Scales : Nails, Inc.

Nails, Inc. (Posted on 2004-07-06)

At Nails, Inc., there is a wide selection of nails. The nails are indistinguishable except for their weights.

This afternoon, you received 4 boxes full of nails. They were labeled A, B, C, and D. You were told that two of the labels were switched on accident and that you must find out which ones.

You have several scales with which you can weigh nails against each other. The people here are picky about efficiency, and you'd like to do all the weighings at once, using as few scales as possible. This means you can't change your weighing strategy according to the results of the first weighing.

What is the smallest number of scales needed to figure out which nails were switched if:

You know that the order from lightest to heaviest is A, B, C, then D, or
You know that an A nail is 1.9 g, B is 2.0 g, C is 2.1 g, D is 2.2 g.

Prove that it is the fewest number of scales needed.

Submitted by Tristan

Rating: 2.7500 (4 votes)

Solution: (Hide)

1. 3 scales
Weigh A against B, B against C, and C against D.
If A›B, then the switched labels were A and B, A and C, or A and D.
If A‹B, then the switched labels were B and C, B and D, or C and D.
If B›C, then the switched labels were A and C, B and C, or B and D.
If B‹C, then the switched labels were A and B, A and D, or C and D.
If C›D, then the switched labels were A and D, B and D, or C and D.
If C‹D, then the switched labels were A and B, A and C, or B and C.
You can use deductive reasoning to find which were switched from these results.
3 is the least number of weighings because there are 6 possible combinations of switched labels. Each scale can go either left or right, so there are 2^3 possible results. You can't expect the nails to balance anytime without knowing their exact weight. Less scales would have less possible results than the possible solutions.
2. 2 scales
Weigh A and D against B and C, and on another scale weigh B against C.
If A+D=B+C, then the switched labels were A and D or B and C.
If A+D‹B+C, then the switched labels were B and D or C and D.
If A+D›B+C, then the switched labels were A and B or A and C.
If B›C, then the switched labels were A and C, B and C, or B and D.
If B‹C, then the switched labels were A and B, A and D, or C and D.
Again, deductive reasoning will give you the answer.
2 is the least number of scales because there are 3^2 possible results with two weighings. Each scale can go left, right, or be balanced. This is enough to determine which of the 6 possible combinations of labels were switched.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle	Marlayna	2004-09-27 14:18:59
re: No Subject	Steve Royer	2004-07-07 23:55:40
re: Sing4	Sing4TheDay	2004-07-07 10:23:01
Sing4	David	2004-07-07 01:12:11
solution	Charlie	2004-07-06 13:42:00
No Subject	Sing4TheDay	2004-07-06 13:30:08

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