You are given 12 seemingly identical metal balls, but one of them weighs slightly more or less than the others. With a balance scale and three weighings only, how can it be determined which is the oddball and whether it is heavier or lighter?

This is one those puzzles that'll likely take a fair amount of time to work out, but the solution (at least the only one I know of) is ingenious and very satisfying to discover. No special math or logic skills required, just perseverance and insight.

Solution to Twelve Balls puzzle

This one requires a little work and inspiration to solve. I know of only this one solution here. If anyone knows of another, I'd be happy to see it.

Divide the balls into three groups of four each; label these groups AAAA, BBBB and CCCC.

Weigh AAAA_BBBB. The possible results are:
  1. If they balance: One of the C's is heavy or light. Therefore:
    weigh CCC_AAA (remember, all A's are now known to be standard):
    1. If they balance: The 4th C is the oddball. Therefore, weigh the 4th C against any other ball.
      1. If the 4th C falls: The 4th C is heavy.
      2. If the 4th C rises: The 4th C is light.
    2. If the CCC side falls: One of the C's is heavy. (Remember, the A's are known to be standard.) Therefore, weigh C_C:
      1. If they balance: The other C is heavy.
      2. If one side falls: The fallen C is heavy.
    3. If the CCC side rises: One of the C's is light. Therefore, weigh C_C.
      1. If they balance: The other C is light.
      2. If one side rises: The risen C is light.
  2. If the AAAA side falls: The oddball is either a heavy A or a light B and the C's are all standard. Therefore, arrange the balls into three new groups like so: AAAC BBBA CCCB. (This re-arrangement step, and the one like it below in step 3, are the key to solving this puzzle.) Weigh BBBA_CCCB:
    1. If they balance: The oddball is in AAAC. Therefore, weigh A_A.
      1. If they balance: The other A in AAAC is heavy.
      2. If one side falls: The fallen side has the heavy A.
    2. If the left side (BBBA) falls: The A in BBBA is heavy or the B in CCCB is light. Therefore, weigh A_C (C is known to be standard).
      1. If they balance: The B in CCCB is light.
      2. If the A side falls: A is heavy.
      3. If the C falls: Not possible.
    3. If the right side (CCCB) falls: The a B in BBBA is light. Therefore, from the BBBA group weigh B_B.
      1. If they balance: The other B in BBBA is light.
      2. If the left side falls: The B on the right is light.
      3. If the right side falls: The B on the left is light.
  3. If the BBBB side falls: The oddball is either a heavy B or a light A and the C's are all standard. Therefore, arrange the balls into three new groups like so: AAAB BBBC CCCA. Weigh AAAB_CCCA:
    1. If they balance: The Oddball is in BBBC. Therefore, weigh B_B.
      1. If they balance: The other B in BBBC is heavy.
      2. If one side falls: The fallen side has the heavy B.
    2. If the left side (AAAB) falls: The B in AAAB is heavy or the A in CCCA is light. Therefore, weigh B_C (C is known to be standard).
      1. If they balance: The A in CCCA is light.
      2. If the B side falls: B is heavy.
      3. If the C side falls: Not possible.
    3. If the right side (CCCA) falls: An A in AAAB is light. Therefore, from AAAB weigh A_A.
      1. If they balance: The other A in AAAB is light.
      2. If the left side falls: The A on the right is light.
      3. If the right side falls: The A on the left is light.