Twelve coins: solution

Statement: Twelve coins.

This puzzle is more difficult than the Twelve marbles puzzle so it requires additional notation. Each weighing can have one of three results: the scale leans to the left (L), the scale leans to the right (R), or the scale stays even (E). Three weighings with three possible results each creates a tree of 3³ = 27 total possibilities:

An answer to this puzzle includes a weighing for each of the 13 times the tree of possibilities above branches. An answer is valid if, for all 27 leaves of the tree, we know which coin is fake and whether it is lighter or heavier than the real coins.

Let’s get started.

The first weighing#

We begin with no information about any of the individual coins’ weights.

If we compare two groups of coins, the scale will tell us which group is heavier. We can start by comparing four of the coins to four others, with the remaining four to the side:

If the fake coin is off the scale, then it will stay even. If the fake coin is on the scale, then it will lean to one side, either left or right. So we have three cases:

L	E	R

Solving case E#

E is the simplest case to solve, so let’s analyse it first. The scale stays even so all coins on it must be real. We can rule them out as suspects:

For our second weighing, let’s compare three of the remaining suspects with three coins that we know are real:

Again, there are three possibilities:

EL	EE	ER

Solving case EL#

Let’s start with case EL. Since the scale leaned to one side, there must be a fake coin on the scale. We can rule out the coin left off the scale: it must be real. We also know that the fake coin must be lighter than real coins; otherwise, the scale would have leaned the other way. So each of the three remaining suspects is either real or lighter, but cannot be heavier:

Because of this symmetry, solving the EL case also solves the ER case.

To solve the EL case, we compare one suspect to another, with the third suspect off the scale:

Again, there are three possibilities:

ELL	ELE	ELR

Let’s start with case ELL. The scale leaned to one side so we know that the fake coin is on the scale. In the previous weighing, we ruled out the possibility of the fake coin being heavier than real coins. This means that the lighter coin must be the fake coin:

Symetrically, for case ELR the lighter coin must be the fake coin:

For case ELE, the fake coin cannot be on the scale. So the coin left off the scale must be fake, and we know it is lighter:

That solves case EL entirely.

Solving case ER#

Cases EL and ER are symmetrical. This is case EL:

And this is case ER:

Because of this symettry, the method to solve case EL, when mirrored, also solves case ER.

Solving case EE#

Case EE was this result to the second weighing:

Since the scale stays even, all coins on it must be real. By elimination, the fake coin must be the suspect we left off the scale:

We use the third weighing to determine whether the fake coin is lighter or heavier than a real coin:

The scale cannot stay even with the fake coin on it, so case EEE is not possible. That leaves case EEL:

And case EER:

In both cases, we know wether the fake coin is lighter or heavier than real coins. This solves case EE and, since we solved cases EL and ER previously, all of case E.

Solving case L#

Case L was this result for the first weighing:

The scale leaned to one side, so the fake coin must be on it. If the fake coin is on the left, then it must be heavier. If the fake coin is on the right, then it must be lighter:

In order to obtain as much information as possible from the LL, LE, and LR cases, we compare a mix of real coins, potentially lighter coins, and potentially heavier coins:

There are three possible results:

LL	LE	LR

Solving case LL#

Let’s start with case LL. Since the scale leans to the left, we know the left side is heavier than the right side. The fake coin must be on the scale, so we can rule out all coins off the scale as suspects.

The coin tagged + on the right side cannot be responsible: the previous weighing ruled out the possibility of it being lighter than a real coin. Similarly, the coin tagged - on the left side cannot be reponsible: we already ruled out the possibility of it being heavier than a real. So we know that both of these coins are real:

This is equivalent to case ER and case EL. We solve case LL in the exact same way.

Solving case LR#

Case LR was this result to the second weighing:

The scale leans to the right, so the right side is heavier than the left side. The fake coin must be on the scale, so we can rule out all coins off the scale as suspects.

The coins tagged + on the left side cannot be responsible: the previous weighing ruled out the possibility of them being lighter than real coins. The only suspects that remain are the coin tagged - on the left and the coin tagged + on the right:

We compare one of these coins to a real one:

The scale cannot lean to the left, since we know the coin on the left side cannot be heavier than a real coin. That leaves two possible outcomes:

LRE	LRR

In case LRE, the scale stays even so we know all coins on it are real. By elimination, the remaining suspect must be the fake coin and we know it is heavier than a real coin:

In case LRR, the only suspect on the scale must be the fake coin and we know it is lighter than a real coin:

That concludes the LR case.

Solving case LE#

Case LE was this result for the second weighing:

The scale stays even so we know all coins on it are real:

This is equivalent to case EL. We solve case LE in the exact same way.

This solves case L entirely.

Solving case R#

Cases L and R are symmetrical: both are solved in the same way, only mirrored. So we have already solved case R.

With cases L, E, and R all solved, we have solved the puzzle.