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Two false coins (Posted on 2005-03-25) Difficulty: 3 of 5
You have ten coins, but two are fake, and weigh a little less. How many times do you have to use a two arm scale, in order to pick out the two fakes?

See The Solution Submitted by Federico Kereki    
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Solution Solution without knowing if the fakes are equal | Comment 10 of 11 |

This solution does not make any assumptions that the fake coins are equal or not equal.

Label the coins A-J.  Weigh them in three groups (AB/CD), (DE/FG), and (GH/IJ).  There are 21 possible cases.
         (AB/CD) (DE/FG) (GH/IJ)
Case 1  :   =       =       =
Case 2  :   =       =       >
Case 2b :   <       =       =
Case 3  :   =       =       <
Case 3b :   >       =       =
Case 4  :   =       >       =
Case 4b :   =       <       =
Case 5  :   =       >       >
Case 5b :   <       <       =
Case 6  :   =       >       <
Case 6b :   >       <       =
Case 7  :   =       <       >
Case 7b :   <       >       =
Case 8  :   =       <       <
Case 8b :   >       >       =
Case 9  :   >       =       >
Case 9b :   <       =       <
Case 10 :   >       =       <
Case 11 :   <       =       >
Case 12 :   >       <       <
Case 12b:   >       >       <

Case 1  :   =       =       =
There are three possibilites for the fakes: C and one of A or B; E and F; H and one of I or J.  Weigh C vs H.  If C is greater, then H is one fake and weigh I vs J to find the other.  If H is greater, then C is one fake and weigh A vs B to find the other.  If C equals H, then E and F are fakes.
Five weighings

Case 2  :   =       =       >
Two of H, I and J are fakes.  Use two weighings to determine which two are fakes.
Five weighings

Case 2b :   <       =       =
Case 2b is the same as Case 2 with the coins labeled in reverse order

Case 3  :   =       =       <
Either E and G are the fakes or H and one of I and J are the fakes.  Weigh I vs J.  If I is greater, then the fakes are H and J.  If J is greater, then the fakes are H and I.  If I and J are equal then E and G are the fakes.
Four weighings.

Case 3b :   >       =       =
Case 3b is the same as Case 3 with the coins labeled in reverse order

Case 4  :   =       >       =
Either E and F are the fakes or G and one of I and J are the fakes.  Weigh I vs J.  If I is greater, then the fakes are G and J.  If J is greater, then the fakes are G and I.  If I and J are equal then E and F are the fakes.
Four weighings.

Case 4b :   =       <       =
Case 4b is the same as Case 4 with the coins labeled in reverse order

Case 5  :   =       >       >
One of F and G is a fake, and one of I and J is a fake.  Use two more weighings to determine which ones are the fakes.
Five weighings

Case 5b :   <       <       =
Case 5b is the same as Case 5 with the coins labeled in reverse order

Case 6  :   =       >       <
Either F and H are fakes or G and one of {E,F,H,I,J} are the fakes.  Weigh the pair EF against the pair IJ.  If the EF equals IJ then G and H are the fakes.  If EF is greater, then G and one if I and J are the fakes, weigh I vs J to find which.  If EF is less, then the fakes are E and G, F and G, or F and H.  Weigh E against H.  If E equals H, then the fakes are F and G; if E is greater, then F and H are the fakes; if H is greater, then E and G are the fakes.
Five weighings.

Case 6b :   >       <       =
Case 6b is the same as Case 6 with the coins labeled in reverse order

Case 7  :   =       <       >
E is one of the fakes.  Either I or J is the other fake.  Use one weighing to determine which is the other fake.
Four weighings

Case 7b :   <       >       =
Case 7b is the same as Case 7 with the coins labeled in reverse order

Case 8  :   =       <       <
E is one of the fakes.  Either G or H is the other fake.  Use one weighing to determine which is the other fake.
Four weighings

Case 8b :   >       >       =
Case 8b is the same as Case 8 with the coins labeled in reverse order

Case 9  :   >       =       >
C is one of the fakes.  Either I or J is the other fake.  Use one weighing to determine which is the other fake.
Four weighings

Case 9b :   <       =       <
Case 9b is the same as Case 9 with the coins labeled in reverse order

Case 10 :   >       =       <
Either D and G are the fakes or C and H are the fakes.  Use one weighing to determine which pair are the fakes.
Four weighings

Case 11 :   <       =       >
One of A and B is a fake, and one of I and J is a fake.  Use two more weighings to determine which ones are the fakes.
Five weighings

Case 12 :   >       <       <
D and G are the fakes
Three weighings

Case 12b:   >       >       <
Case 12b is the same as Case 12 with the coins labeled in reverse order

In any case, it takes at most five weighings to separate two light fakes without knowing wether or not the fakes are equal.


  Posted by Brian Smith on 2005-06-02 21:18:14
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