The fake coin problem

Should be seven times. Weigh two coins at a time, so that's six times. One of those will produce an imbalance. Take one of the coins from the imbalanced result and weigh it against a different coin. If there is an imbalance again, that's the fake coin. If it's balanced, the other coin from the imbalance is the fake.

Put 6 coins on each side of the scale. One side will be lighter, and one will be heavier. Take all the coins off, and put 3 on each side from the set of 6 coins that were on the lighter side. If the piles weigh the same, then the fake is heavier than normal. If they're different, then the fake is lighter than normal.

If the fake is lighter than normal, take the lighter of the 2 piles you just weighed. Compare any 2 of those 3 on the scale. If they weigh the same, the third one is the fake. If they're different, the lighter one is the fake.

If you found out the fake is heavier than normal, then split the heavier pile from the original weighing into piles of 3 and compare them. Take the heavier of the 2, and weigh any 2 of those 3. If they weigh the same, the third one is the fake. If they're different, the heavier one is the fake.

Thats three weighings if the fake is lighter than normal, and four if the fake is heavier than normal.

ABCDEF vs GHIJKL (1)

one will turn up heavier, say its the abc group

ABC vs DEF (2)

if one doesn't turn up heavier, then weigh other set

GHI vs JKL (3)

one of these will turn up lighter, weigh each of them 1 by 1 against a coin from the ABCDEF set, whichever turns up lighter is correct

J v A (4)

K v A (5)

L v A (6)

EDIT strikeout part was stupid but yeah up until then i had theyango's method

max by this method is 6

Actually, max is 4. Read TheYango's post to find out why.

Split it up into two groups of six. Weight it. One group will be heavier than the other. (A-F) (G-L) (#1)

Take one group of six. Split it into two groups of three. If they are identical weight, then use the other group of six for the next step. If they are not of identical weight, then use this group for the next step. (#2) Say that (A-C) (D-F) are of identical weight, and (G-I) and (J-L) are not.

Weight (G-I) against (A-C). (#3) If they are equal weight, then use the group (J-L) for the next step. If not, then use the group (G-I) for the next step.

Choose two random coins (say G and H if G-I was used from the previous step) from that group and weigh them from each other. (#4) If they are equal, then the one that remains (I) is the odd one out. If not, then weigh one of them (say, G) against coin A. (#5) If they are equal, then the other one (H) is the odd one out. If they are not, then G is the odd one out.

Steps 4-5 can probably be optimized though. =/

Coins A-L.
Weigh ABCD v EFGH and ABCD v IJKL.
If ABCD is the same as either, the other 4 contain the fake and you know if the fake is lighter/heavier by whether that pile is lighter/heavier (e.g. if ABCD = EFGH and ABCD > IJKL, then IJKL has the fake and the fake is lighter). If ABCD is different, then ABCD contains the fake, and you know if the fake is lighter/heaver by whether ABCD is lighter/heaver than the other two.

Call the pile with the fake now PQRS.
Weigh PQ v RS, choose the one you know has the fake from whether the fake is heavier/lighter as determined above, and weigh those. The fake is lighter/heavier as determined above.

3 weighs every time.
EDIT: wow, I can't count. Will figure out in a bit.

(ABCDEF) versus (GHIJKL)
(ABCGHI) versus (DEFJKL)

Find which group of three is heavier or lighter.

Take two from the group of three, measure. If equal, odd one out. If not equal, you know whether the odd one is supposed to be lighter/heavier and that one is it.