This article was originally published in the Summer/Autumn 2015 edition of the MTCircular. The activity was also facilitated by Chris Bolognese during the Math Teachers’ Circle Network Virtual Workshop in 2020.
People who wonder about mathematical objects and ideas see math not as the quest for The Answer, but as an opportunity to play and discover. Humans are wired to think in this manner. This sense of wonder is what motivates a question regarding a “perfect ruler” which was recently explored at our Columbus Math Teachers’ Circle:
Is it possible to measure all possible integer lengths on a ruler without marking every integer on that ruler? In particular, can you construct the most efficient ruler that can measure all integer lengths from 1 inch to 36 inches on a yardstick using the least number of marks? And if so, what is the minimum number of marks needed and where should they be placed?
A trivial solution would be to mark the ruler at one inch and simply measure objects by moving the ruler along the object one inch at a time. So, we constrain this exploration to using the ruler without moving it along the object.
Observations
As is often the case, it may be easiest to begin the exploration with a simpler example. Many of our Circle’s participants chose to start by analyzing a 6-inch ruler such as the one shown in Figure 1. Take a moment to strategize about how you might make this ruler most efficient.
Participants primarily used guess-and-check to remove marks at various locations. For instance, the mark at 2 inches could be removed, since an object of length 2 inches can be still be measured, say, as the difference in length between the marks at 1 inch and 3 inches. Additionally, we can remove the mark at 5 inches, since we can still measure a length of 5 inches from the 1-inch mark to the right end. If we remove the mark at 3 inches, we can still measure a length of 3 inches between the 1-inch and 4-inch mark (and a length of 2 inches can still be measured between the 4-inch mark and the right end of the ruler). Since we cannot do even better with just one mark, a most efficient ruler of length 6 inches requires only two marks (Figure 2).
Conjectures
With a successful exploration in hand for the 6-inch ruler, some groups advanced to longer ruler lengths, while others reverted back to analyze even simpler rulers of shorter lengths to look for a cumulative pattern. After additional time to explore, groups shared their conjectures, including but not limited to the following:
- A most efficient ruler must have at least one mark either 1 inch from the left or right end, in order to measure one less than the ruler’s length.
- Symmetry about the midpoint of the ruler allows for twice the amount of possible solutions. For example, marks could be placed at 2 inches and 5 inches, instead of 1 inch and 4 inches, to create another most efficient 6-inch ruler.
- Marks at the Triangular Numbers (1, 3, 6, 10, 15, 21, etc.) might be an efficient marking scheme. So might marks at the Fibonacci Numbers (1, 2, 3, 5, 8, 13, 21, etc.).
- Marks at consecutive integers are not an efficient marking scheme.
Delving Deeper
While the original task of finding the most efficient yardstick was not exclusively answered by our Circle, a lot of beautiful mathematics unfolded in the process. To investigate this situation further, two digital tools were created after the session by Math Circle members. One tool was written in Javascript to allow the user to choose a length and the location of marks (Figure 3). The tool keeps track of which lengths are measurable and which still cannot be measured. See if you can find any patterns between the length of the ruler and the number of minimum marks required. It is free to use at http://gadgets.mathplusacademy.com/ruler/ruler.html.
Leveraging computer science even farther, we wrote open-source Ruby code to compute the most efficient rulers of various lengths. Visit it online at https://ideone.com/rd0OzG. One interesting result from this code is the minimum number of marks as a function of length, graphed in Figure 4 at above right. Note that this includes marking schemes that are mirror images of each other. We feel that this graph most closely resembles the graph of a square root function. Do you see any other graphical relationships?
Conclusions
“Perfect rulers” proved to be an engaging and challenging problem for all. It fit the criteria for a “rich mathematical task”:
- Easy to understand with a low barrier to entry
- Offers opportunities for initial success to promote engagement
- An “open middle” that allows for creativity and multiple strategies and tools
- Encourages collaboration and discussion
- Easily extendable with “what if” and “what if not” questions
It is interesting to note that the original “perfect ruler” problem could not be solved by any group during the Circle meeting. It is precisely this fact that led several participants to pursue their investigation of this problem beyond the meeting. It is a reminder to teachers at all levels that it is often beneficial to allow rich mathematical problems to remain unsolved because it inspires mathematical thinking outside the classroom. We encourage your Circle to explore this problem at length!
The Authors
Chris Bolognese (bolognesechris@gmail.com) is a mathematics teacher and Department Chair for Columbus Academy Upper School and is a co-founder of the Columbus Mathematics Teachers’ Circle. Raj Shah (raj@mathplusacademy.com) is the founder of Math Plus Academy©, a business for K-8 students to excel in STEM learning, and is a regular participant and presenter at the Columbus Mathematics Teachers’ Circle. Find more information about our Circle at https://columbusmathcircle.wordpress.com/. Raj was introduced to this problem by Matt Enlow, who teaches Upper School Mathematics at the Dana Hall School in Boston, Mass.