# One False Coin amongst 12

talzamir
Posers and Puzzles 25 Sep '11 08:55
1. talzamir
Art, not a Toil
25 Sep '11 08:55
You have 12 coins. One of the 12 differs from the others by weight, not much but enough that a balance scale can differentiate it from the others, but you don't know whether it is heavier and lighter than the others.

It is easy enough to find out which coin is heavier or lighter than the others, and whether it is heavier or lighter, by using the scales four times.

Can you do it with only three uses of the balance scale?
2. 26 Sep '11 12:554 edits
What do we have to pay each time we use these scales?

Yes,
I'll number the coins from 1-12

First test:

1 2 3 4 vs 5 6 7 8
If no response the fake is coins 9, 10, 11 or 12 which can be found out in two turns by:

9 10 vs 1 2
and
9 vs 11
(this is the easy case) If the first reading is unbalanced we use the fact the we know now that 9-12 are not fake and choose out next try as:

1 2 5 9 vs 3 4 7 10
(If the scales are balanced the fake is either 6 or 8 which we can determine by 6 vs 12)
If the scales give the same result as in test 1, the fake coin is either 1,2 or 7.
If we get a different result (i.e. the scales flip) the fake is 3,4 or 5. Let's say for example the result is the same (1,2 or 7) then for our final test we do

1 7 vs 11 12
If it's balanced the fake is 2, the same result as in tests 1 and 2, coin 1 is the fake and if the results change again it's number 7. (Exactly the same principle with 3,4 or 5 -- 3 5 vs 11 12).
3. 27 Sep '11 20:55
You're not finding out whether the fake coin is heavier or lighter with this algorithm. Not sure what the real answer is yet, not that easy.
4. talzamir
Art, not a Toil
27 Sep '11 22:041 edit
SOLUTION

Number the coins from 1-12, as above.

First weighing: coins 1, 2, 3, 4 vs 5, 6, 7, 8

If there are in balance:

Second weighing: 1 2, 3 vs 9, 10, 11

If they are in balance, coin 12 is false,

and the third weighing tells

whether it&#039;s heavy or light.

If they are not, it tells that

one of the coins 9..11 is false

AND it tells whether&#039;s heavy or light.

So the third weighing is 9 vs 10, finding the false coin.

If the original 1..4 vs 5..8 does not give balance:

Second weighing: 2..5 vs 1, 9, 10, 11

If the scale tilts the same was as it did,

the false coin is one of coins 2..4, and

we know whether it&#039;s heavy or light.

The final weighing is 2 vs 3.

If the scale now tilts the opposite way,

the false coin is coin 1 or coin 5,

and for both we know whether it&#039;s

heavy or light if it is false.

1 vs 2 as last weighing gives the

rest of the information.

If the scale is now balanced,

the false coin is one of coins 6..8

and we know whether it&#039;s heavy or light.

The third weighing, 6 vs 7,

gives the last missing bit of information.