The 12 coins and the scales

1. Case C: in weighing 1 lowers the right (symmetrical of B)
2. The real hypotheses are 24: if it falls 12: 12 is heavy.
3. Each weighing has 3 possible results: left low, balance, right low. With 3 weighings there are up to 3^3=27 possible patterns.

Answer: Yes, you can identify the fake coin and decide if it is heavier or lighter by exactly 3 weights.
1) Information framework (why 3 weighings are enough)
Each weighing has 3 possible results: left low, balance, right low.
With 3 weighings there is up to

$$ 3^3=27 $$

possible patterns.
The real hypotheses are 24:

• 12 fake currency options,
• and for each one, two types: heavy or light.

Like $27>24$, a well-designed strategy can distinguish all cases.
2) First weighing (main partition)
Heavy 1:

$$ (1,2,3,4)\,\text{vs}\,(5,6,7,8). $$

3 branches open.
Case A: equilibrium
So the false one is in $\{9,10,11,12\}$.
Heavy 2:

$$ (9,10,11)\,\text{vs}\,(1,2,3) $$

(where 1,2,3 are good).

• If it balances: the false one is 12.

Weighing 3: $12$ vs $1$.

• if it goes down 12: 12 is heavy;
• if it goes up 12: 12 is light.
• If the $9,10,11$ side goes down: the fake one is one of those and it is heavy.

Weighing 3: $9$ vs $10$:

• if it goes down 9: 9 heavy;
• if it drops 10: 10 heavy;
• if they balance: 11 heavy.
• If the $9,10,11$ side goes up: the fake one is one of those and it is light.

Weighing 3: $9$ vs $10$:

• if it goes up 9: 9 light;
• if it goes up 10: 10 light;- if they balance: 11 light.

Case B: in heavy 1 the left goes down
Candidates:

• weighed in $\{1,2,3,4\}$, or
• light in $\{5,6,7,8\}$.

Heavy 2:

$$ (1,2,5)\,\text{vs}\,(3,6,9) $$

with 9 good.

• If balanced: candidates $\{4\text{ pesada},7\text{ ligera},8\text{ ligera}\}$.

Weighing 3: $7$ vs $8$:

• if they balance: 4 heavy;
• if it drops 7: 8 light;
• if it drops 8: 7 lightly.
• If lower left: candidates $\{1\text{ pesada},2\text{ pesada},6\text{ ligera}\}$.

Weighing 3: $1$ vs $2$:

• if they balance: 6 light;
• if it goes down 1:1 heavy;
• if it goes down 2: 2 heavy.
• If you go down right: candidates $\{3\text{ pesada},5\text{ ligera}\}$.

Weighing 3: $3$ vs $9$:

• if they balance: 5 light;
• if it goes down 3: 3 heavy.

Case C: in weighing 1 lowers the right (symmetrical of B)
Candidates:

• weighed in $\{5,6,7,8\}$, or
• light in $\{1,2,3,4\}$.

Heavy 2:

$$ (5,6,1)\,\text{vs}\,(7,2,9) $$

with 9 good.

• If it balances: candidates $\{8\text{ pesada},3\text{ ligera},4\text{ ligera}\}$.

Weighing 3: $3$ vs $4$:

• if they balance: 8 heavy;
• if it drops 3: 4 light;
• if it drops 4: 3 lightly.
• If lower left: candidates $\{5\text{ pesada},6\text{ pesada},2\text{ ligera}\}$.

Weighing 3: $5$ vs $6$:

• if they balance: 2 light;
• if it goes down 5: 5 heavy;
• if it goes down 6: 6 heavy.
• If you go down right: candidates $\{7\text{ pesada},1\text{ ligera}\}$.Weighing 3: $7$ vs $9$:
• if they balance: 1 light;
• if it goes down 7: 7 heavy.

3) Methodological conclusion
It is not about memorizing a table: it is about building a decision tree where each result reduces the set of hypotheses without ambiguity. This strategy separates the 24 hypotheses into 3 levels and always ends with unique identification.