# Which coins are Counterfeit or Real?

Ok i recieved this problem and i have been stuck on this. Heres the problem : There are 12 identical coins, 1 is counterfeit. This counterfeit can be either LIGHTER or HEAVIER than the real coins. You can use a balance scale ONLY 3 TIMES. Find the counterfeit coins and determine whether it is lighter or heavier……All you have to do is find steps on how you did it and come up with an answer to see if it can be lighter or heavier, You can only use the balance 3 times. Thanks

Heres the thing, you can only use the scale to help you, no telescopes or using your hands

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1. Among twelve similar coins one is counterfeit. It is not known whether the counterfeit coin is lighter or heavier that the genuine coin (but all the genuine coins weigh the same). Using at most three weighings on balance scales, find the counterfeit coin and determine whether it is lighter or heavier than a genuine coin.

Solution: Put 4 coins in each pan of the scales.

1a. If the scales balance, then the remaining 4 coins contain the counterfeit coin. Call is “suspicious group” and number coins in this suspicious group 1,2,3,4.

2a. Place coins 1,2,3 on one pan of the scales and 3 genuine coins taken from any of the coins used in step 1a that balanced, as we now know those must be genuine). If the scales balance, coin 4 is counterfeit. Use the third weighing to compare coin 4 to one of the genuine ones to figure out whether it is lighter or heavier. Otherwise one of coins 1,2,3 is counterfeit.

3a. (If here, then one of coins 1,2,3 is counterfeit and 2 weighings have been used). Note in this case you already know from 2a whether the fake coin is heavier or lighter because 1,2,3 were weighed again known genuine coins. Assume without loss of generality that the fake coin is lighter. Weigh coins 1 and 2. If they balance, then coin 3 is the fake one. Otherwise, the lighter one of coins 1 and 2 is the fake one.

1b. If here, one group of 4 used in the first weighing is heavier than the other (assume without loss of generality left is heavier). Then the remaining group has only genuine coins. Name coins in the heavier group 1h, 2h, 3h,4h, and the one in the lighter group 1l, 2l, 3l and 4l.

2b. Compare 1h,2h, 3l and 3h, 4h, 4l. If they balance, then the counterfeit coin is either 1l or 2l, and the counterfeit coin is lighter. Use the third weighing to compare 1l and 2l, the lighter one of which is then the counterfeit one.

3b. (If here the scale did not balance in step 2b).

Case 3b.1: 1h,2h,3l is heavier. In this case 3h, 4h and 3l are genuine. To see that note that if either 3h or 4h were counterfeit, then in step 2b the group 1h,2h,3l would be lighter, which is not the case. Likewise, if 3l were the counterfeit coin, then 1h,2h,3l would be the lighter group in step 2b, which is not the case. This means that the counterfeit coin is one of 1h, 2h and 4l. Compare 1h, 4l to two genuine coins (e.g. those in the “good” group determined in the first weighing). If the scale balances, then 2h is the counterfeit, and it is heavier. If the “good” group of 2 coins is heavier, then 4l is the counterfeit, and it is lighter. Otherwise, if the “good” group of 2 coins is lighter, then 1h is the counterfeit, and it is heavier.

Case 3b.2: 1h,2h,3l is heavier. This is exactly analogous to case 3b.1.

Cant you use your hands to weigh the coins??/

Split the 12 coins into groups of three (four in each group).

Compare the 1st group to the second group. (record if they are balanced and record the weight, if applicable).

Compare The second group to the third group (record if they are balanced and record the weight, if applicable).

Compare The third group to the 1st group. (record if they are balanced and record the weight, if applicable).

And which group is either light or heavier is the group with the counterfeit coin.

From there, you examine with a telescope to determine which coin is different than the rest.

Good luck, and I’m pretty sure this will work, if this is for class, I hope that you can use a telescope.

Good luck again!