# I got asked this one in a job interview. It cooked my noodle for several minutes.?

You have 12 coins, one of which is counterfeit, and a balance pan scale. The fake coin may be identified by the fact that it’s weight is different from the 11 genuine coins. Can you identify the counterfeit coin and whether it is heavier or lighter in three weighings?

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Yes, you place a pile of 6 coins on either side of the scales, which ever set of six is lighter you split into 2 sets of 3 coins and do the same, then with the lightest set of 3 coins, you weigh 2 of them, if they weigh the same, the coin you didn’t weigh is the counterfeit. Or which ever of the 2 coins was the lightest.

Yup - split the coins into 2 piles of 6, put these on the scale. Then you will see which pile is heaver so you can remove the lighter ones, do this for 2 piles of 3 as well. once you have 3 remaining you will be able to put 1 and 1 on the pan, if one is heaver, its the fake - if they are both the same then the remaining coin is the fake. WHALA!!

Not unless you know that the fake is definitely heavier or definitely lighter. You could do it in 3 weighs then, but I think you need four to establish the truth if you are just told the weight is different.

You place six on each side. Then remove the heavier six coins. Next, split those six into three on each side of the balance. Again remove the heavier three. Now you put two of the remainning three on the balance (one on each side). You can determine which of the three is fake. If one of the two on the balance is lighter that is the fake, otherwise, it is the remaining coin.

no

You need to number the coins 1-12. Then weigh them in this sequence.

1 4 6 8 v 7 10 11 12

2 4 9 11 v 8 5 10 12

7 10 6 11 v 3 5 9 12

1st weigh unbalance only. coin = no.1 (L=heavy R=light)

2nd weigh unbalance only. coin = no.2 (L=heavy R=light)

3rd weigh unbalance only. coin = no.3 (R=heavy L=light)

1st&2nd unbalance same way. coin = no.4 (L=heavy R=light)

2nd&3rd unbalance same way. coin = no.5 (R=heavy L=light)

1st&3rd unbalance same way. coin = no.6 (L=heavy R=light)

1st&3rd unbalance opp. way. coin = no.7 (on 3rd L=heavy R=light)

1st&2nd unbalance opp. way. coin = no.8 (on 1st L=heavy R=light)

2nd&3rd unbalance opp. way. coin = no.9 (on 3rd R=heavy L=light)

1st2nd&3rd unbalance 3rd opp. way. coin = no.10 (on 3rd L=heavy R=light)

1st2nd&3rd unbalance 1st opp. way. coin = no.11 (on 3rd L=heavy R=light)

1st2nd&3rd unbalance same way. coin = no.12 (R=heavy L=light)

(L=balance is tilted left and R = balance tilted right)

Step 1:

Weigh 4 coins on each side of the scale.

scenario A - balance -> counterfeit coin is amongst the other 4 unused coins. (step 2)Take any 3 unused coins vs 3 used coins.

if balance -> (step 3.1)last coin is counterfeit so just weigh it vs any coin to determine if its heavier or lighter.

if not -> (step 3.2) you’d know the counterfeit coin is heavier or lighter depending on whether it rises(lighter) or sinks(heavier). take any 2 and weigh them vs each other to determine the counterfeit coin.

scenario B is when Step 1 does not balance. We’ll label the coins for easier illustration. The side that rises L-lighter, sinks H-heavier and unused coins N-normal.

Step 2:

LLLHH vs LNNNN. If balance, (step 3.1) just weigh the remaining 2 Hs to determine which is the heavier counterfeit coin.

If LNNNN rises -> either one of HH (in LLLHH) is heavier or L (in LNNNN) is lighter. (step 3.2) weigh the 2 Hs vs each other to determine if one of them is the heavier counterfeit coin or the L (in LNNNN) is the lighter counterfeit coin.

If LLLHH rises -> one of the 3 Ls is the lighter counterfeit coin. (step 3.3) weigh any 2 Ls vs each other to determine which is the lighter counterfeit coin.

It’s a bit lengthy but I dunno how else to simplify it. Hope you can understand my explanation.

Cheers