Imagine three doors where behind one door $\text{A}$ there is a new car, behind door $\text{B}$ there is a goat, and behind door $\text{C}$ there is a new car and a goat.
The problem is that each door is labeled incorrectly…
If you can open only one door, is it possible to label all the doors correctly?
I'm trying to answer this in logical form, but I am unsure if my answer is good enough:
Denote doors as $D_1, D_2, D_3$
$D_n $ is equal to either $A,B$ or $C$.
If we assume that the correct solution is $$D_1 = A
$$
$$D_2 = B
$$
$$D_3 = C
$$
then
$$\begin{align}
D_3 = A &\implies D_1=B \implies D_2 = C\\
D_3 = B &\implies D_1=C \implies D_2 = A\\
\end{align}$$
$$\Leftrightarrow$$
$$\begin{align}
C = D_1 \implies D_2 = A \implies D_3 = B\\
C = D_2 \implies D_1 = B \implies D_3 = A\\
\end{align}$$
Would you consider this a valid solution?
Best Answer
The key is the assertion that each door is labelled incorrectly. If capitals stand for the correct labels and lower case letters for the incorrect ones currently present, you will find there are just two possibilities:
$$Ab: Bc :Ca$$
$$Ac :Ba :Cb$$
Each of the doors is labelled differently in the two cases. Whichever door you open, you will therefore know which case applies, and will be able to apply correct labels without opening a second door.