The height shelf

1. The fourth note is 3: that already completely fixes the fourth book.
2. Once the fourth is set, the fifth note also greatly restricts the last position.
3. Then all that remains is to order three heights so that they produce 0, 1 and 1.

Answer:

$$ 5,\ 2,\ 4,\ 1,\ 3. $$ Let's call the heights from left to right $a_1,\dots,a_5$. - The fourth note is $3$. Since there are only three positions to the left of the fourth book, those three have to be higher than him. Then $$

a_4=1.
$$ - The fifth note is $2$. We already know that $a_4=1$, so among the four books on his left there must be exactly two taller than him. This forces the last book to be

$$ a_5=3, $$

because if it were $2$ it would have three higher ones on the left, and if it were $4$ or $5$ it would have less than two. The heights $2,4,5$ remain for the first three positions, with notes $0,1,1$. - The first note is $0$, so the first book has to be the highest of those three: $a_1=5$.

• For the second and third notes to both be $1$, the remaining two positions can only be $2$ and $4$ in this order. Therefore, the complete row is:

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