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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.

  Submitted by Brian Smith    
Rating: 3.5556 (9 votes)
Solution: (Hide)
Choose two weights to compare for the first weighing. Choose two weights other than the first two to compare for the second weighing. There are three possible cases:
Case 1:Both weighings are equal
Case 2:One weighing is equal and one is not
Case 3:Both weighings are not equal

Case 1:Both weighings are equal
Choose one weight from weighing 1 and one weight from weighing two and weigh them. There are two possible results: they are equal or they are not equal

- Subcase 1: the third weighing is equal
Label the weights so that the weighings were A=B, B=C, C=D and E is the last weight. Since this is a broken scale and none of the weights have the same weight, the weights are in consecutive order of size, either A>B>C>D or D>C>B>A. Weigh A vs D to find out which is the correct order. Then weigh C vs E to find out which end E belongs.
Total weighings: 5

- Subcase 2: the third weighing is not equal
Label the weights so that the weighings were A=B, C=D, A>C and E is the last weight. A,B,C and D must be one of the three orders: A>B>D>C, B>A>D>C, or A>B>C>D. Weigh (C+A) vs (D+B). If there is a difference, that difference is at least 2g so the broken scale will be able to sense it. If (C+A)=(D+B) then the order is A>B>D>C. If (C+A)>(D+B) then the order is A>B>C>D. If (D+B)>(C+A) then the order is B>A>D>C. Relabel A,B,C,D so that the inequality is W>X>Y>Z. Weigh E vs X. If X>E then weigh E vs Y to determine if the order is W>X>Y>E>Z or W>X>Y>Z>E. If X=E then weigh E vs Y to determine if the order is W>X>E>Y>Z or W>E>X>Y>Z. If E>X then the order is E>W>X>Y>Z.
Total weighings: 5

Case 2:One weighing is equal and one is not
Label the weights so that the weighings were A>B, D=E and C is the last weight. Weigh (A+B) vs (C+D). There are three possible results: (A+B)>(C+D), (A+B)=(C+D) and (C+D)>(A+B).

- Subcase 1: (A+B)>(C+D)
A is the heaviest (A=55). B and C are consecutive and C is not consecutive to D or E. C is either consecutive to A or the smallest (C=54 or 51). Weigh A vs C and B vs D. If A=C and B=D then the order is A>C>B>D>E. If A=C and D>B then the order is A>C>B>E>D. If A>C and B=D then the order is A>E>D>B>C. If A>C and D>B then the order is A>D>E>B>C.
Total weighings: 5

- Subcase 2: (A+B)=(C+D)
A is the heaviest and B is the lightest (A=55 and B=51). C is not the middle weight (C!=53). Weigh A vs C and C vs D. There are four possible results. If A=C and C=D then the order is A>C>D>E>B. If A=C and C>D then the order is A>C>E>D>B. If A>C and C=D then the order is A>E>D>C>B. If A>C and D>C then the order is A>D>E>C>B.
Total weighings: 5

- Subcase 3: (C+D)>(A+B)
B is the lightest (B=51). A and C are consecutive and C is not consecutive to D or E. C is either consecutive to B or the largest (C=55 or 52). Weigh B vs C and A vs D. If B=C and A=D then the order is E>D>A>C>B. If B=C and A>D then the order is D>E>A>C>B. If C>B and A=D then the order is C>A>D>E>B. If C>B and A>D then the order is C>A>E>D>B.
Total weighings: 5

Case 3:Both weighings are not equal
Take the two weights from the first weighing as a group and take the two weights from the second weighing as a group. Weigh those two groups of two. There are two possible results: they are equal or they are not equal.

- Subcase 1: the third weighing is not equal
Label the weights so that the weighings were W>Y, X>Z, (W+Y)>(X+Z), and V is the last weight. W,X,Y and Z must be in the order W>X>Y>Z. Weigh V vs X. If X>V then weigh V vs Y to determine if the order is W>X>Y>V>Z or W>X>Y>Z>V. If X=V then weigh V vs Y to determine if the order is W>X>V>Y>Z or W>V>X>Y>Z. If V>X then the order is V>W>X>Y>Z.
Total weighings: 5

- Subcase 2:the third weighing is equal
Label the weights so that the weighings were A>B, C>D, (A+B)=(C+D), and E is the last weight. E must be the middle weight (E=53). The order of the weights is either A>C>E>D>B or C>A>E>B>D. Weigh A vs E. If A=E then the order is C>A>E>B>D, if A>E then the order is A>C>E>D>B.
Total weighings: 4

In every case, at most five weighings were made. It is impossible to solve by making at most four weighings. There are 120 possible ways for the five weights to be mixed up at the beginning but since the scale has only three outputs, only 3^4=81 different cases could be distinguished, and that is not enough.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Can be done in only 4 stepsManish2005-03-21 06:46:28
re: Inferior alternativeVivek2005-03-08 13:09:36
Hints/Tipsre: 5 weighings is enoughDetti2004-10-06 09:38:12
Hints/Tips5 weighings is enoughDetti2004-10-06 08:47:14
ANOTHER PUZZLEdebbie2004-09-28 19:19:49
No Subjecti_wish2004-09-11 18:01:58
Hints/TipsRe: 5 weighings--REALLY!!!i_wish2004-09-11 13:05:01
re: 5 weighingsCharlie2004-09-08 15:44:02
5 weighingsi_wish2004-09-07 18:17:18
Some ThoughtsMy Poor Brainbob9092004-09-06 14:12:59
Some ThoughtsNo Subjectktii2004-08-30 04:43:21
Some Thoughtsre: Inferior alternativedhruv2004-08-20 23:35:33
re: The rest of my commentCharlie2004-08-03 11:41:38
re: Inferior alternativeCharlie2004-08-03 08:52:50
Inferior alternativePenny2004-08-03 06:14:07
The rest of my commentCharlie2004-08-02 16:12:29
Some ThoughtsDon't know if this is the fewest weighings...Charlie2004-08-02 16:08:36
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