Three. (Full marks 50 points) On a straight ruler of length $36 \mathrm{~cm}$, mark $n$ graduations so that the ruler can measure any integer $\mathrm{cm}$ length in the range $[1,36]$ in one go. Find the minimum value of $n$.


Three, if the ruler is marked with 7 (or fewer than 7) graduations, we can prove that it is impossible to measure any integer length in the range [1, 6] cm in a single measurement.

In fact, 7 graduations, including the two end lines of the ruler, total 9 lines, which have 36 different combinations. Therefore, 7 graduations can measure at most 36 different lengths.
If there are two segments of the same length among the 8 segments created by 7 graduations, the number of different lengths that can be measured will be less than 36, clearly failing to meet the measurement requirement.

If none of the 8 segments created by 7 graduations have the same length, note that \(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36\). It is easy to see that at this point, there is one segment of each length from \(1 \mathrm{~cm}\) to \(8 \mathrm{~cm}\).
\(1^{\circ}\) If the segment of length \(1 \mathrm{~cm}\) is not at one end of the ruler, then it cannot measure \(35 \mathrm{~cm}\); if the segment of length \(2 \mathrm{~cm}\) is not at one end of the ruler, then it cannot measure \(34 \mathrm{~cm}\).
\(2^{\circ}\) If the segment of length \(1 \mathrm{~cm}\) is at one end of the ruler and the segment of length \(2 \mathrm{~cm}\) is at the other end, then if the segment of length \(3 \mathrm{~cm}\) is not adjacent to the segment of length \(1 \mathrm{~cm}\), it cannot measure \(32 \mathrm{~cm}\); if the segment of length \(3 \mathrm{~cm}\) is adjacent to the segment of length \(1 \mathrm{~cm}\), it cannot measure \(31 \mathrm{~cm}\).

From this, it is evident that to meet the measurement requirement, the number of graduations on the ruler must be at least 8.
8 graduations divide the ruler into 9 segments. When the lengths of these 9 segments are \(1, 2, 3, 7, 7, 7, 4, 4, 1\), it is known that it can achieve the measurement of any integer length in the range \([1, 36]\) cm in a single measurement.

Therefore, the conclusion of this problem is: the minimum value of \(n\) is 8, meaning that at least 8 graduations must be marked on a 36 cm ruler to possibly achieve (the given graduation distribution is one method) the measurement of any integer length in the range \([1, 36]\) cm in a single measurement.
(Contributor: Liu Yue, Tongling No. 3 Middle School, Anhui Province, 244000)