[Guest]

You are presented with a set of 12 suspect coins. Of these, it

is known that the number of counterfeit coins is either 0, 1 or 2. All of

the counterfeit coins, if any, are known to have come from a single batch

of light coins. You must identify the counterfeit coins, if any, after 4

weighings or less.

Describe the weighing possibilities to determine which coin(s) are fake if any.

Edited July 19, 2011 by njBLOB

[CaptainEd]

These problems with up to two bad coins are a real eye-opener for me: I keep thinking I ought to be able to invent the next weighing, and I keep being wrong. Thank you for the educational opportunity.

The chart contains 79 rows, one per case.

A case is represented by 12 weights: L denotes a light coin, and x denotes an ordinary coin. Each case has 0, 1, or 2 light coins.

The coin weights are presented in order, assuming that the coins have been numbered 123456789ABC.

A weighing is presented by using an "x" to separate the coins in the left pan from those in the right pan. The other coins are not part of the weighing.

The first weighing is shown under column W1 (for Weighing 1), and the results are shown in column R1 (for Results 1).

Of course, all the cases were subjected to the same W1. The results divided into three cohorts, two of size 26 and one of size 27, shown by the breakdown in R1.

Now break down one of those cohorts with a second weighing. Notice that the W2 for the H cohort is not the same as the W2 for the x cohort. Observe the results, R2.

Continue in this way. Each cohort consisting of cases that have the same weighing history and the same result have to be given the same next weighing. However, other cohorts may or may not use the same next weighing.

I hope this is clear enough. If not, please ask.

Coins W1 R1 W2 R2 W3 R3 W4 R4

xxxxxxxLxxxL 1234x5678 H 159Bx26AC H 7x8 H AxC H

xxxxxxxLxLxx 1234x5678 H 159Bx26AC H 7x8 H AxC L

xxxxxLxLxxxx 1234x5678 H 159Bx26AC H 7x8 H AxC x

xxxxxxLxxxxL 1234x5678 H 159Bx26AC H 7x8 L AxC H

xxxxxxLxxLxx 1234x5678 H 159Bx26AC H 7x8 L AxC L

xxxxxLLxxxxx 1234x5678 H 159Bx26AC H 7x8 L AxC x

xxxxxLxxxxxL 1234x5678 H 159Bx26AC H 7x8 x AxC H

xxxxxLxxxLxx 1234x5678 H 159Bx26AC H 7x8 x AxC L

xxxxxLxxxxxx 1234x5678 H 159Bx26AC H 7x8 x AxC x

xxxxxxxLxxLx 1234x5678 H 159Bx26AC L 7x8 H 9xB H

xxxxxxxLLxxx 1234x5678 H 159Bx26AC L 7x8 H 9xB L

xxxxLxxLxxxx 1234x5678 H 159Bx26AC L 7x8 H 9xB x

xxxxxxLxxxLx 1234x5678 H 159Bx26AC L 7x8 L 9xB H

xxxxxxLxLxxx 1234x5678 H 159Bx26AC L 7x8 L 9xB L

xxxxLxLxxxxx 1234x5678 H 159Bx26AC L 7x8 L 9xB x

xxxxLxxxxxLx 1234x5678 H 159Bx26AC L 7x8 x 9xB H

xxxxLxxxLxxx 1234x5678 H 159Bx26AC L 7x8 x 9xB L

xxxxLxxxxxxx 1234x5678 H 159Bx26AC L 7x8 x 9xB x

xxxxxxxLxxxx 1234x5678 H 159Bx26AC x 45x78 H 7x8 H

xxxxxxLxxxxx 1234x5678 H 159Bx26AC x 45x78 H 7x8 L

xxxxxxLLxxxx 1234x5678 H 159Bx26AC x 45x78 H 7x8 x

xxxxLxxxxxxL 1234x5678 H 159Bx26AC x 45x78 L AxC H

xxxxLxxxxLxx 1234x5678 H 159Bx26AC x 45x78 L AxC L

xxxxLLxxxxxx 1234x5678 H 159Bx26AC x 45x78 L AxC x

xxxxxLxxxxLx 1234x5678 H 159Bx26AC x 45x78 x BxC L

xxxxxLxxLxxx 1234x5678 H 159Bx26AC x 45x78 x BxC x

xxxLxxxxxxxL 1234x5678 L 159Bx26AC H 3x4 H AxC H

xxxLxxxxxLxx 1234x5678 L 159Bx26AC H 3x4 H AxC L

xLxLxxxxxxxx 1234x5678 L 159Bx26AC H 3x4 H AxC x

xxLxxxxxxxxL 1234x5678 L 159Bx26AC H 3x4 L AxC H

xxLxxxxxxLxx 1234x5678 L 159Bx26AC H 3x4 L AxC L

xLLxxxxxxxxx 1234x5678 L 159Bx26AC H 3x4 L AxC x

xLxxxxxxxxxL 1234x5678 L 159Bx26AC H 3x4 x AxC H

xLxxxxxxxLxx 1234x5678 L 159Bx26AC H 3x4 x AxC L

xLxxxxxxxxxx 1234x5678 L 159Bx26AC H 3x4 x AxC x

xxxLxxxxxxLx 1234x5678 L 159Bx26AC L 3x4 H 9xB H

xxxLxxxxLxxx 1234x5678 L 159Bx26AC L 3x4 H 9xB L

LxxLxxxxxxxx 1234x5678 L 159Bx26AC L 3x4 H 9xB x

xxLxxxxxxxLx 1234x5678 L 159Bx26AC L 3x4 L 9xB H

xxLxxxxxLxxx 1234x5678 L 159Bx26AC L 3x4 L 9xB L

LxLxxxxxxxxx 1234x5678 L 159Bx26AC L 3x4 L 9xB x

LxxxxxxxxxLx 1234x5678 L 159Bx26AC L 3x4 x 9xB H

LxxxxxxxLxxx 1234x5678 L 159Bx26AC L 3x4 x 9xB L

Lxxxxxxxxxxx 1234x5678 L 159Bx26AC L 3x4 x 9xB x

xxxLxxxxxxxx 1234x5678 L 159Bx26AC x 15x34 H 3x4 H

xxLxxxxxxxxx 1234x5678 L 159Bx26AC x 15x34 H 3x4 L

xxLLxxxxxxxx 1234x5678 L 159Bx26AC x 15x34 H 3x4 x

LxxxxxxxxxxL 1234x5678 L 159Bx26AC x 15x34 L AxC H

LxxxxxxxxLxx 1234x5678 L 159Bx26AC x 15x34 L AxC L

LLxxxxxxxxxx 1234x5678 L 159Bx26AC x 15x34 L AxC x

xLxxxxxxLxxx 1234x5678 L 159Bx26AC x 15x34 x 9xA L

xLxxxxxxxxLx 1234x5678 L 159Bx26AC x 15x34 x 9xA x

xxxLxxLxxxxx 1234x5678 x 1259x346A H 3x4 H 6x7 H

xxxLxLxxxxxx 1234x5678 x 1259x346A H 3x4 H 6x7 L

xxxLxxxLxxxx 1234x5678 x 1259x346A H 3x4 H 6x7 x

xxLxxxLxxxxx 1234x5678 x 1259x346A H 3x4 L 6x7 H

xxLxxLxxxxxx 1234x5678 x 1259x346A H 3x4 L 6x7 L

xxLxxxxLxxxx 1234x5678 x 1259x346A H 3x4 L 6x7 x

xxxxxxxxxLxL 1234x5678 x 1259x346A H 3x4 x BxC H

xxxxxxxxxLLx 1234x5678 x 1259x346A H 3x4 x BxC L

xxxxxxxxxLxx 1234x5678 x 1259x346A H 3x4 x BxC x

xLxxxxLxxxxx 1234x5678 x 1259x346A L 1x2 H 5x7 H

xLxxLxxxxxxx 1234x5678 x 1259x346A L 1x2 H 5x7 L

xLxxxxxLxxxx 1234x5678 x 1259x346A L 1x2 H 5x7 x

LxxxxxLxxxxx 1234x5678 x 1259x346A L 1x2 L 5x7 H

LxxxLxxxxxxx 1234x5678 x 1259x346A L 1x2 L 5x7 L

LxxxxxxLxxxx 1234x5678 x 1259x346A L 1x2 L 5x7 x

xxxxxxxxLxxL 1234x5678 x 1259x346A L 1x2 x BxC H

xxxxxxxxLxLx 1234x5678 x 1259x346A L 1x2 x BxC L

xxxxxxxxLxxx 1234x5678 x 1259x346A L 1x2 x BxC x

xxxxxxxxxxLL 1234x5678 x 1259x346A x 123x7AB H 9xC H

xxxxxxxxLLxx 1234x5678 x 1259x346A x 123x7AB H 9xC L

xxxxxxxxxxLx 1234x5678 x 1259x346A x 123x7AB H 9xC x

xLxxxLxxxxxx 1234x5678 x 1259x346A x 123x7AB L 1x2 H

LxxxxLxxxxxx 1234x5678 x 1259x346A x 123x7AB L 1x2 L

xxLxLxxxxxxx 1234x5678 x 1259x346A x 123x7AB L 1x2 x

xxxxxxxxxxxL 1234x5678 x 1259x346A x 123x7AB x 4xC H

xxxLLxxxxxxx 1234x5678 x 1259x346A x 123x7AB x 4xC L

xxxxxxxxxxxx 1234x5678 x 1259x346A x 123x7AB x 4xC x

Edited August 7, 2011 by CaptainEd