(Puzzle number 21 in Boris Kordemsky’s Moscow Puzzles)
I belong to a group of teachers, mathematicians, and puzzle developers who advocate the use of math-based puzzles in the K-12 classroom. To this end, every April for over a decade we have held a weekend math fair workshop at BIRS. (BIRS = Banff International Research Station). Although BIRS exists to aid research in mathematics and related disciplines, from its inception it has also supported educational initiatives, and our April workshops are an example of its continuing commitment.Erica doesn't like odd numbers, so the box of chocolates shown above meets with her approval. The problem is that she has to remove six chocolates from the box in such a way that she leaves an even number of chocolates in each row and each column.Make a 4 by 4 grid, and using pennies or other tokens as chocolates, show how she can do this. There is more than one solution.
About 20 participants attend the workshops, many of them teachers and mathematicians who share a common interest in enhancing mathematics education. The participants are diverse in background and experience, and the workshops have quite a wide scope. Although the emphasis is on the use of mathematical puzzles and games, the freedom and informality of the workshops allows the discussion to veer off in related directions. For example, when teachers illustrate how they fit puzzles into their teaching, their presentations sometimes spark a discussion on how they have adapted to various other aspects of the math curriculum.
At every workshop, we promote the use of a non-competitive puzzle-based math fair to elevate the students interest in mathematics. In a math fair of this kind, the students are in charge of booths which present puzzles to passers-by. The students are not there to demonstrate the solutions, but rather to give hints and suggestions to help the visitors solve the puzzle. The whole affair is very interactive.
A visitor to the math fair might encounter the puzzle at the top of this post, where the students provide a 4 by 4 grid and sixteen removable “chocolates” for the visitors to work with. The students presenting the puzzle would have previously solved it themselves (without the help of parents or guardians) and would have mastered and practiced at least one solution. The students would also be expected to recognize whether or not a visitor has a solution. Depending upon the grade level, students could even be expected to answer a visitor who asks “Is zero an even number?” or “Do I have to worry about the diagonals?” In other words, the students will have become experts on the evensies puzzle.
If you have tried the evensies puzzle you will know that is not instantaneously solvable. Most people will attack the puzzle using a “trial and error” or a “guess and check” approach—this is the way I first solved it, and, in fact, it is the way most of the participants at one of the recent workshops solved it. However, there are other ways to tackle the puzzle, and asking older students to find an approach that does not depend exclusively on trial and error would be a way to ramp up the mathematics involved. Perhaps one way to invoke a discussion about this would be to ask an entire class to solve the puzzle independently and display the different solutions they have obtained, and then ask them what they they notice.
Perhaps someone will know a way to prompt the students to ask “What do you mean when you say two solutions are the same?” This is an important question but it may be difficult to answer because understanding “sameness” depends not only the context but also upon the student’s mathematical background. For example, most people agree that the numbers 1/8 and 0.125 are the same but many will balk when told that this is also true for the numbers 1.0 and 0.9999… (where the nines go on forever). With regard to the evensies puzzle, a grade 6 student would not be expected to understand “sameness” in the same way as a grade 12 student.
The evensies puzzle involves parity (the properties of even and odd numbers). Even before solving the puzzle, students may have learned a few facts like “even + even is even” and “odd + even is odd”. The puzzle offers a meaningful context for a deeper investigation. One could ask students if it is possible to remove 5 chocolates and still have an even number in each row and column. Or if it is possible to remove a certain number of chocolates and end up with an odd number in each row and column. Or one could ask students to create a similar puzzle for a 3 by 3, or a 5 by 5, or a 6 by 6 grid.
Some teachers have come to our math fair workshops specifically because they have either heard about or visited a puzzle-based math fair. Others have come because they have already used puzzles in the classroom, and are willing to share their knowledge. And some have come because they have learned about the workshops from other teachers. Whatever the reasons, feedback from the teachers has always been very positive.
Some mathematicians have come to the workshop because they teach a math course for Education students who, as part of their course, will be presenting a math fair to local schools. Some mathematicians come because they are interested in K-12 education and will be involved in some way with their school district. If a mathematician is fond of recreational math puzzles, and if he or she is interested in K-12 education, the workshop is a great setting to merge both interests. In addition they will almost certainly encounter some math puzzles and games that they have not seen before, and that’s always interesting.
The feedback from the mathematicians has also been positive. Some have told me that they were impressed by the quality and enthusiasm of the teachers at the workshops. They say that observing the teachers has made them reflect on their own teaching and assessment methods, and that it has provided ideas on how to help their own students take ownership of their learning.
For some resources and information about non-competitive puzzle-based math fairs, I would urge you to visit the SNAP math fair website. If you want more information about the math fair workshop, please feel free to contact me at firstname.lastname@example.org.