Beautiful problem - Finding the fake coin of 5 in 3 weighings?

Question

You have 5 coins. One of them is a fake (different weight). It is unknown whether it is heavier or lighter. You have an electronic scale which displays the weight of the coins which you put on it, rather than a balance scale which compares weights. You must determine which coin is the fake in only 3 weighings.

I know the answer, but the problem is so nice I must put it out to you guys.

The scale is NOT a balance scale. Read more carefully =P

There's another solution, slightly more complicated. See if you can get it ;)

Answer 1

I'll give it a shot,
Divide the 5 coins into 3 groups:
Group 1: 2 coins
Group 2: 2 coins
Group 3: 1 coin

Step 1:
Weighing 1: Measure Group 2, weight = a
Weighing 2: Measure Group 3, weight = x

If a = 2x then x must be the weight of the correct coins and the fake must be in Group 1. Otherwise skip to Step 3.

Step 2:
Weighing 3: Measure one coin from Group 1.
If it equals x, then the other coin is fake. Otherwise the coin you measured is fake.


Step 3:
If after Weighing 2 'a' isn't equal to 2x then either Group 2 or 3 has the fake coin.

Weighing 3: Measure one coin from Group 2
3 Possibilities now:
1) If this weight is equal to x, then the coin you didn't measure in Group 2 Is fake.
2) If, however, the weight = a / 2, then the Group 3 coin is fake.
3) If neither the above is true, then the coin you just measured is fake.


Hope it's correct!

Answer 2

I understand from the question that it is enough to determine which is the fake. No need to determine heavier or lighter:

Let's label coins: ABCDE

1. Weigh A

2. Weigh B and C together.

If B+C = 2A, then the fake is D or E, so...

3a. Weigh A and D together.

If A + D = 2A, then E is fake. Otherwise, D is fake.

If B + C > 2A, then the fake is either A (light) or B or C (heavy)

3b. Weigh A and B together

If A + B = 2A, then C is fake
If A + B > 2A, then B is fake
If A + B < 2A, then A is fake

Similar solution if B + C < 2A (reverse the conclusions of the last 2)

Answer 3

1st weighing:
Put 2 coins on each side. If they are equal, the unweighed coin is the fake. If the 2 sides are not equal, go to step 2.

2nd weighing: Replace 1 coin on the scale by the 5th coin from above. If the 2 sides are now equal, the coin you just replaced is the fake. If not, go to step 3.

3rd weighing: Replace a coin from the other side of the scale with the coin you left out in step 2. If the sides are now equal, the coin you replaced is the fake. If not, the coin you did not replace is the fake. Either way, you have found it.

Answer 4

when 3 coins are indivisually placed inside a fully filled bucket of water,the amount of water displaced will be different for fake and original coins.