The height shelf

1. Everything fits together, and the solution is unique.
2. Taking $a_1=4$, $a_2=2$, $a_3=3$: for $a_1=4$, there are 0 higher on the left.
3. Taking $a_1=4$, $a_2=2$, $a_3=3$: for $a_2=2$, only 4 is higher (1).

Back to problem
Answer:

$$ 4,\ 2,\ 3,\ 1,\ 5. $$

Let the row be $a_1,\dots,a_5$.

• Note 5 = 0: there is no one taller to your left $\Rightarrow a_5=5$.
• Note 4 = 3: your 3 on the left are higher $\Rightarrow a_4=1$.

$\{2,3,4\}$ remains for $a_1,a_2,a_3$ with notes $0,1,1$.
Taking $a_1=4,\ a_2=2,\ a_3=3$:

• for $a_1=4$, there are 0 higher ones on the left,
• for $a_2=2$, only $4$ is higher (1),
• for $a_3=3$, only $4$ is higher (1).

Everything fits together, and the solution is unique.