In Fall 2019, our Advanced Geometry class officially published a new integer sequence in the Online Encyclopedia of Integer Sequences (OEIS). We recently spoke with Derek Graves, Math and Computer Science Instructor about this achievement and what it means.
DG: Please allow me to provide the background and context for why this is such an incredible achievement. The OEIS is the definitive record of integer sequences for mathematicians. An integer sequence is an ordered set of integers that counts something or many different things. The Fibonacci sequence is one that you may have heard of. Publishing a new sequence in the OEIS is considered an achievement for professional mathematicians and carries with it a mark of pride. Few undergraduates publish sequences, let alone middle and high school students, so this is quite a feat.
TGS: How did the discovery of a new, unpublished sequence come about?
DG: We began the year by attempting to solve a problem about amoebas and blankets. The prompt was as follows: You have a pet amoeba that lives on a dot grid. Your amoeba always takes the form of a closed shape with boundaries along the grid, like so:
Your amoeba is cold and desperately needs a blanket, but it is very picky. It only likes perfectly square blankets whose corners all lie on dots of the grid on the boundary of the amoeba, like the orange square here:
Does there exist an amoeba for which no proper blanket can be found? This is actually an open question in the field of mathematics—mathematicians themselves don’t know the answer.
TGS: I assume your lesson had an expectation of teaching the students how to stay resilient through failure — what was their process?
DG: Students began their collaborative problem-solving efforts with guess-and-check. They constructed amoebas and tried to find proper blankets for those amoebas. As a class, they quickly realized that guess-and-check is an inefficient solution method for this problem. It may eventually yield a solution, but it could take forever and is not guaranteed to find an amoeba with no blankets, even if one exists.
As they thought more deeply about the problem, the class came up with a brilliant idea: instead of creating amoebas and trying to find blankets that fit them, what if we enumerated all possible blankets and constructed an amoeba that “misses” at least one corner of every blanket. That would give us our desired blanket-less amoeba.
In trying to enumerate all possible squares, they discovered their integer sequence. The students realized that in order to count blankets, they had to specify the size of the dot grid. First, they let the grid be 3×3, then 4×4, 5×5, etc. At each size, they counted the number of squares. This list of total number of squares for each dot grid size is their sequence. The official link is here: A328152 (“328152” is the catalogue number of their sequence within the OEIS). We submitted the sequence on October 5, and it was published October 21, 2019.
TGS: It must be very fulfilling as a teacher to see your students achieve this honor.
DG: It’s fantastic. I am incredibly proud of their collaborative problem-solving approach that enabled them to find and publish new mathematics.
Derek Graves
teaches upper-level mathematics courses and electives at The Grayson School. His experience includes developing math curriculum — both daily instruction and enrichment programming for gifted middle- and upper-school students. Mr. Graves is particularly interested in helping support students’ ability to communicate their understanding of math effectively. Read more about how Derek makes learning experiential and memorable in his classroom.
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