Discovering New Math and Hidden Talents at Our Public Puzzle Board

written by: Dr. Tony McCaffrey

Discovering New Math and Hidden Talents at Our Public Puzzle Board

The Puzzle Board

Outside my classroom in the new PJM STEM center is a wall of whiteboards that I keep filled with puzzles of all types: spatial, visual, verbal, mathematical, logic, insight (i.e., ones in which people usually experience an aha moment), and divergent (i.e., ones in which there are an unlimited number of solutions). Called simply the Puzzle Board, it has become a gathering spot next to the cafe for working on puzzles. Students meet other students and faculty there who share a love for puzzles of various types.
The puzzles are written in blue marker and students put up their answers in red. If the answer is correct, it is erased but the student’s initials are left in place. In this way, other students can work on a puzzle from scratch while the initials of the solvers remain to announce who was clever enough to figure it out. Occasionally, someone forgets to leave their initials and we have a Good Will Hunting situation on our hands of figuring out who is the mysterious problem solver.
At all times, the Puzzle Board is a place of pure enjoyment because students work on these puzzles of their own choosing and not because anything was assigned to them.
A New Math Discovery
Occasionally, a new math discovery happens at the Puzzle Board, as happened this year in the month of November. Two students worked on the following problem together.
When added, 2 and 2 equal 4, and when multiplied 2 and 2 also equal 4. What are other numbers that give the same answer when added or multiplied?
One student immediately thought out of the box: the solutions could involve more than two numbers. This student noticed that the collection 3, 2, 1 produced 6 both when the numbers are multiplied and when they are added. The other student then noticed that there are an infinite number of solutions: 4, 2, 1, 1 will work; 4, 3, 1, 1, 1, 1, 1 will work; and on and on. The first student then wrote up a formula to calculate the structure of any solution. Soon, they were looking into negative numbers, then fractions, and then the real numbers.
Very quickly, we were writing up a short, two-page math paper showing this elegant formula and this strange relationship between multiplication and addition. A series of Google searches suggested that this may be an original discovery, so we submitted the paper to, which is a site for pre-prints of professional math and science papers so new ideas can spread quickly. We will keep you informed what the community has to say.
Discovering Special Talents
Some students at the Puzzle Board learn for the first time that they have a special talent they did not know they had. Visual pattern recognition and spatial thinking are rarely required in school subjects, so if a student makes a self-discovery, it is usually one of these two types.
Through puzzles, many Eagle Hill students already have discovered an exceptional talent they did not know they had. Almost all of the self-discoveries written about were for pattern recognition and spatial thinking, but a couple were for extraordinary skills in problem solving. Some students are highly gifted in their problem solving abilities. Given a situation, they can figure out what is important about it and what math might be useful. In school, this skill generally remains hidden because in their school work they usually are told beforehand exactly what aspects are important and what math to use.
In sum, our Puzzle Board is a fun “watering hole” of the mind where new discoveries happen occasionally and self-discoveries happen regularly. Students who like the same types of puzzles can find each other and, through their initials, students showcase their cleverness to the whole school community. And all this happens while sipping a great drink from the neighboring cafe, the more literal “watering hole” of our school.

Dr. McCaffrey teaches Computer Science and is the author of Infinite Learning Diversity: Uncovering the Hidden Talents of Our Students (Rowman & Littlefield, 2019)
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