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Yet one more coin sorting problem (Posted on 2004-05-31) Difficulty: 3 of 5
You have five coins, apparently alike, but actually of different weights. You also have a two arm scale.

Can you manage to sort the coins in ascending order, using the scale only seven times?

Bonus question: can it be done in fewer weighings?

See The Solution Submitted by Federico Kereki    
Rating: 4.0000 (4 votes)

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Solution | Comment 9 of 20 |
Choose two weights for group 1 and choose another two weights for group 2. Compare the two group 1 weights against each other and compare the two group 2 weights against each other. Compare the heavier of group 1 against group 2.

After these three weighings, relabel the weights as follows: Label the weights such that A>B, A>C>D, and E is unknown. A and B are from one of the groups, C and D are from the other and E is the weight in neither group.

These first three weighings reduced the possible orderings from 120 to 15, and the following weighings continue to divide the possibilities in half. The next two weighings find the position of E in ACD, and the final two weighings find the position of B in the last four positions.

Weigh E against C:
Case 1: E>C
 Weigh A against E:
 Case 1.1 E>A
  Weigh B against C
  Case 1.1.1 B>C
   The order is EABCD
  Case 1.1.2 C>B
   Weigh B against D
   If B>D then the order is EACBD
   If D>B then the order is EACDB
 Case 1.2 A>E
  Weigh B against C
  Case 1.2.1 B>C
   Weigh B against E
   If B>E then the order is ABECD
   If E>B then the order is AEBCD
  Case 1.2.2 C>B
   Weigh B against D
   If B>D then the order is AECBD
   If D>B then the order is AECDB
Case 2: C>E
 Weigh E against D
 Case 2.1 E>D
  Weigh B against E
  Case 2.1.1 B>E
   Weigh B against C
   If B>C then the order is ABCED
   If C>B then the order is ACBED
  Case 2.1.2 E>B
   Weigh B against D
   If B>D then the order is ACEBD
   If D>B then the order is ACEDB
 Case 2.2 D>E
  Weigh B against D
  Case 2.2.1 B>D
   Weigh B against C
   If B>C then the order is ABCDE
   If C>B then the order is ACBDE
  Case 2.2.2 D>B
   Weigh B against E
   If B>E then the order is ACDBE
   If E>B then the order is ACDEB

  Posted by Brian Smith on 2004-05-31 15:22:06
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