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Five Weights and a Broken Scale (Posted on 2004-08-02) Difficulty: 4 of 5
You have to sort five weights weighing 51g, 52g, 53g, 54g, and 55g. You have a balance scale with which you can compare the weights. But after solving so many sorting puzzles, it is starting to break down.

If the difference between two weights is greater than 1.5g, the scale will correctly determine which side is heavier. If the difference between the weights is less than 1.5g or equal, the scale will indicate the weights are equal.

Sort the weights in the smallest number of weighings.

See The Solution Submitted by Brian Smith    
Rating: 3.5556 (9 votes)

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5 weighings | Comment 9 of 17 |

My answer is 5.

<o:p> </o:p>

Suppose the weights are: A B C D E  (any letter will represent 51,52,53,54,55)

<o:p> </o:p>

I)                   Find out the relation ship between:

A ___ B (>, < or =)

A ___ C

A ___ D

<o:p> </o:p>

The only possible instances you can get are:

a)      >,>,> (A>B,A>C,A>D)

b)      >,>,= (A>B,A>C,A=D)

c)      >,=,=  etc

d)      >,=,<

e)      =,=,<

f)        <,<,<

<o:p> </o:p>

Note: itís impossible to get 3 =ís (pretty obvious)

II)                 <o:p> </o:p>

A) suppose you got case a) >,>,>  

then it follows that A is 55 since it is larger than the three others. It canít be 54 since the balance will display 54=53. This also leaves us to conclude that E is 54!

<o:p> </o:p>

Now we know that A=55, E=54:

<o:p> </o:p>

Compare  B ___ D

            A1) if B>D, then B=53 and D=51  (not one of them can be 52 since the balance will display 51=52 or 52=53) therefore it follows that C=52<o:p></o:p>

In this case, we only needed 4 weighings<o:p></o:p>

            A2) if B=D then B or D must be either (51 and 52) or (52 and 53)

               Compare either B ___ C  or D ___ C

               Letís say you took B ___ C:

<o:p> </o:p>

            If B=C then B=52. why? Because we know that B must only be 51,52, or 53. the only way B=C and B=D at the same time is if itís the middle value, 52.

            Then we are left to think whether of C and D are 51 or 53.

            Compare either C ___ E or C ___ D

            Say you chose C ___ E

            If E>C, it must follow that C =51. It canít be 53 since the scale will display 54=53

            Therefore, we are left to conclude that D=53.

            In this case we had to go up to 5 weighings

The analysis for case a and case f are the same!

<o:p> </o:p>

B) Suppose you got >,>,=   

            A>B, A>C, A=D.

            Then either A=54 or A=55 (think about it! Same reasoning as the previous example)

            B1) Compare E __ C or E __ B

                        Suppose you chose E ___ C


<o:p> </o:p>

Ö Iím too lazy to continue explaining this but if you continue along the same line of reasoning, then answer is 5! (Thatís what I came up with..)

<o:p> </o:p>

Iíll try to write the rest next time if I got more time okay. See yaíll!...

By the way , Iím new here. I love this site!!!! :)

<o:p> </o:p>

  Posted by i_wish on 2004-09-07 18:17:18
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