441. The Broken Ruler. Here is an interesting puzzle that resembles (although in reality, it is significantly different from) one of the classic problems of Bache about a weight cut into pieces, with which it is possible to determine the weight of any load from 1 pound to the total weight of all pieces. In our case, a person has a ruler, the end of which has broken off, so its length is now 33 cm. Most of the divisions on the ruler have worn off, so only 8 of them can be distinguished. Nevertheless, with the help of the ruler, it is possible to measure any whole number of centimeters from 1 to 33.

![](https://cdn.mathpix.com/cropped/2024_05_21_56f37ca27ac3d928838dg-153.jpg?height=146&width=508&top_left_y=692&top_left_x=774)

Where are the remaining divisions located?

For example, I have shown a ruler 13 cm long with four divisions. If I need to measure 4 cm, I measure 1 and 3 cm; if 8 cm, then 6 and 2 cm; if 10 cm, then 3, 1, and 6 cm, and so on. Of course, the necessary measurement should be made by applying the ruler once; otherwise, we could obtain any number of centimeters by sequentially measuring 1 cm, which would deprive the puzzle of any meaning.
441. Let 8 divisions break a 33-centimeter ruler into 9 parts of lengths \(1,3,1,9,2,7,2,6,2\) cm. Then with their help, any whole number of centimeters from 1 to 33 cm can be measured. Of course, the divisions themselves are located at distances of \(1,4,5,14,16,23,25\) and 31 cm from one of the ends of the ruler. Another solution is \(1,1,1,1,6\), \(6,6,6,5\) cm.

This puzzle has at least 16 solutions. I found a rule that can determine the minimum number of divisions for rulers of any length and list some solutions, but a general law that governs all solutions has not yet been found.

[Although a general rule has not been found to this day, significant progress has been noted since Dudeney posed this problem. It has been discovered that eight divisions are also sufficient for a 36 cm ruler. - M. G.]