1. Subscribertalzamir
    Art, not a Toil
    60.13N / 25.01E
    Joined
    19 Sep '11
    Moves
    45043
    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. Joined
    22 Aug '08
    Moves
    9361
    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. Joined
    06 Aug '06
    Moves
    1945
    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. Subscribertalzamir
    Art, not a Toil
    60.13N / 25.01E
    Joined
    19 Sep '11
    Moves
    45043
    27 Sep '11 22:041 edit
    SOLUTION

    Reveal Hidden Content
    Number the coins from 1-12, as above.

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


    Reveal Hidden Content
    If there are in balance:

    Reveal Hidden Content
    Second weighing: 1 2, 3 vs 9, 10, 11


    Reveal Hidden Content
    If they are in balance, coin 12 is false,

    Reveal Hidden Content
    and the third weighing tells

    Reveal Hidden Content
    whether it's heavy or light.

    Reveal Hidden Content
    If they are not, it tells that

    Reveal Hidden Content
    one of the coins 9..11 is false

    Reveal Hidden Content
    AND it tells whether's heavy or light.

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


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


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


    Reveal Hidden Content
    If the scale tilts the same was as it did,

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

    Reveal Hidden Content
    we know whether it's heavy or light.

    Reveal Hidden Content
    The final weighing is 2 vs 3.


    Reveal Hidden Content
    If the scale now tilts the opposite way,

    Reveal Hidden Content
    the false coin is coin 1 or coin 5,

    Reveal Hidden Content
    and for both we know whether it's

    Reveal Hidden Content
    heavy or light if it is false.

    Reveal Hidden Content
    1 vs 2 as last weighing gives the

    Reveal Hidden Content
    rest of the information.


    Reveal Hidden Content
    If the scale is now balanced,

    Reveal Hidden Content
    the false coin is one of coins 6..8

    Reveal Hidden Content
    and we know whether it's heavy or light.

    Reveal Hidden Content
    The third weighing, 6 vs 7,

    Reveal Hidden Content
    gives the last missing bit of information.
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