Burning ropes as timers - How many time intervals can be measured?

Question

^{Note: This is a follow-up question to Burning ropes as timers. The following question and its answers may contain spoilers.}

To sum up the puzzle, these are the rules (slightly modified):

• You have some number N of ropes, each with the following property.
• When you light one end of a rope, the fire will reach the other end after exactly one hour.
• You start the puzzle by lighting one or more ends of the ropes.
• You may light or extinguish any end of a rope after that.
• You can only light or extinguish a rope once another rope has completely burned out.
• You may only light the end of a rope, since lighting anywhere else is inaccurate.
• You may not organize the ropes so that they light each other or themselves.
• You must accurately measure some interval of time, between any two distinct times.

The question is, how many intervals of time (excluding 0) is it possible to measure with N ropes?

For example, when N = 1, the answer is 2.

• 1 hour (by burning the length of the rope)
• 1/2 hour (by lighting both ends of the rope at once)

Answer 1

To get started on this problem, here is a general observation for any $n$:

You can only measure intervals of time of the form $$\frac{m}{2^n}\cdot 1\mbox{h}$$

where $m$ and $n$ are non-negative integers.

Proof (using the extremal principle):

Suppose there was a number of ropes you could use to measure an interval which is not of the proposed form above. Take the minimum number of ropes $n$ for which this is possible, then $n≥2$ (as for $n=1$ it is not possible as shown in the question).

There must be a way of lighting the $n$ ropes so that the interval $\ t = t_2-t_1$ between two "distinct times"¹ $\ t_1$ and $t_2$, where $t_1<t_2$, cannot be described by the above form. As $t_2>0$, there must be a rope $r$ that has completely burned down (otherwise, $t_2$ would not be a distinct time - see the footnote). Due to $t_1<t_2$, you can measure $t_1$ without using $r$ (so $n-1$ ropes suffice), which means that $t_1$ has the form above due to the minimal choice of $n$.

Consider the very last time $t_3$ rope $r$ is lit. By the same argument as above (you do not need $r$ to measure this time), it follows that $t_3$ has the above form. Accordingly, $\left|t_3-t_1\right|$ is also of that form. As $t = t_2-t_1 = (t_2-t_3)+(t_3-t_1)$ is by assumption not of the above form (but $\left|t_3-t_1\right|$ is), it follows that $t_2-t_3$ is not of the form above either.

However, this cannot be true:

Look at amounts of time $t_{r,1},\ldots,t_{r,l}$ that $r$ actually burns. These satisfy the relation $$n_1\cdot t_{r,1} + \ldots + n_1\cdot t_{r,l} = 1\mbox{h}$$

where $n_i = 1,2$, depending on whether $r$ burns at one or both ends. All the amounts of time $t_{r,1},\ldots,t_{r,l-1}$ are of the above form, because they are measured without using $r$. So $t_{r,l}$ also has to be of the form above. But $t_{r,l} = t_2-t_3$, which is a contradiction to the conclusion of the last paragraph.

Therefore, the assumption that there is a number of ropes which can be used to measure an amount of time not of the form above has to be wrong.

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^{1. The start of the experiment or the time a rope has burned out, as there are no other possible events.}