His introductory remarks included mention of how there is a huge emphasis lately on applications. But many aspects of mathematics are beautiful and curious - which can help students “fall in love with mathematics”. To that end, here is a movie (x+y) and a play (The Curious Incident of the Dog In the Night - adapted from a novel) and a book (The Mathematician’s Shiva). And for the musically inclined, a math teacher in Toronto is composing music using math (aleatoric or “chance” music).
Ron’s first encounter with recreational mathematics was magic squares. He said you could do this physically by giving students numbers from 1-9 and having them stand to make a square. There is logic behind this, eg. discovering “5” has to be in the middle since you need to make 15 four different ways through the centre. But while that’s true for 3x3, there are 880 magic square types which are 4x4. Ron also showed an 8x8 magic square which has both SUM (840) and PRODUCT (2,058,068,231,856,000) magic! (Made with help of factoring. See image below.)
He moved on to ambigrams (a symbolic representation that retains meaning when viewed in another orientation), like ZoonooZ, the San Diego zoo newsletter. This happens with numbers/dates like 1961. Scott Kim is a puzzle master who can turn ‘Ignorance’ upside-down to spell ‘Knowledge’. Also Geomagic Squares by Lee Sallows, which don’t use numbers, but shapes or words.
From there, a look at sequences. Pick any number. Count how many letters exist in the English representation. Write that number out, continue. You get deadlocked at FOUR, and you get there REALLY fast - in English. In Spanish, it’s a different cycle. Ron also spoke about “The Kruskal Count” in connection to “Twinkle, Twinkle, Little Star”, hitting the word YOU. Also, the more words you have on a page, the higher the probability it will work to land on a particular word by the end of the page.
There are LOTS of possible rules when you have only three numbers! Ron illustrated with 8, 3, 1/2, using the rule: a, b, (b+1)/a. (Hence 1/2 comes from (3+1)/8, which then becomes the new ‘b’.) That CYCLES after 6 terms. You can plot this using six points (a, b). If a=7 and b=6, you get a right angle! When else does this happen? Consider more extensions: To have the 4th number be 17, what “a” and “b” should he start with?
Then there’s the “Hailstone Sequence”, “another one that is classic recreational mathematics”. Start with a number, x. If even, divide it by 2 (x/2). If odd, multiply by 3 and add 1 (3x+1). Mathematicians believe that the number 1 always makes an appearance, no matter, what you start with (leading to a 1, 4, 2 cycle)... but no one has proved this yet! (An audience member remarked on rounding the logarithm of numbers, to play musical notes.) And poems made using Fibonacci Numbers for syllables are called FIBS. See Greg Pincus for more of those.
Ron then spoke about “Interactive Magic Tricks” which have mathematics as their basis. This can be done in a prerecorded ‘YouTube’ way, and goes back to David Copperfield, “The Orient Express”, inviting people to move fingers on their TV screens. (Before that, Tony Spina and “Room for Doubt”, on a cassette tape in 1950s, and before THAT, Martin Gardner.) Other curiosities include hexa-hexaflexagons (a strip of paper with six sides), the “Four Bug Problem” (with shoutout to Mary Bourassa), and geometrical vanishes (an 8x8 square that seems to become 5x13). This last has a connection to Fibonacci.
|MAGIC SQUARE: SUM AND PRODUCT!|
After Ron’s keynote, we split off into three breakouts (primary, intermediate, senior). I attended the Intermediate one.
Barry started by building off Ron’s work, showing an “Anti-Magic Square”. One where all the totals are not the same, but rather form a sequence of ten CONSECUTIVE numbers, in some order. (So instead of all eight sums being 15, they might be 7,8,9,10,11,12,13,14 in some order.) One of the CEMC (Centre for Education in Mathematics and Computing) problems he had featured one, where to finish a particular square, you needed to determine all the other unknowns first.
The rest of the session was mainly looking at a handout package of sample problems and past contests. Within that, Barry first looked at a problem about distances between pairs of cities. Like the magic square, at what point is the problem “under defined”? (Too many gaps, leading to multiple right answers.) Do you WANT a unique solution? Sequences in particular are very, very dangerous for this. Don’t be afraid to stray from “what the textbook says” if it still makes sense.
Contest-wise, Barry stated that to come up with the multiple choice options, you need to identify the misconceptions. Only one answer there is right, the other solutions (known as “distractors”) come from those misconceptions. How do they decide on which distractors to use? Well, all committee members for a contest submit possible questions in May/June (knowing the correct answer). So when the committee meets in November to come up with first drafts, they may use incorrect answers from other committee members.
That said, if it’s an obvious error that a LOT of the students may make, they DON’T want to use that distractor. They want the student to realize their answer isn’t there (termed the “sucker answer”), think a bit more, and fix the common misconception. I was reminded a bit of Nik Doran’s “Hinge Questions”, where each answer pinpoints a particular misconception to be addressed - and can even allow for partial credit. So this has applications in the regular classroom and/or in creating tests.
Also addressed was what question is better as an open question, as opposed to a multiple choice? Generally, if you could take out the answers, and it would still be good. (So not mere arithmetic.) Also, if there’s a struggle for good “distractors” (without giving things away), then it’s better open. And if it’s possible to get the right answer the wrong way, the question shouldn’t be used at all. It was noted that there are different committees for the different contests.
Consider even a simple question: In a 3x8 grid, how many rectangles are there? This is approached differently depending on grade level. Consider this approach: There are 9 vertical lines, and any unique pair will define a rectangle. Now include the top and bottom, and we can take the product of our two numbers to figure out total rectangles. Noted that in a 4x4 grid there are 30 squares (students answer everything from 17-40), but actually 100 rectangles (and students back off from that, thinking that’s too much).
For solving averages, a +/- approach is possible rather than algebra. Many questions can be done with prime factors. Taking things apart can “simplify” things (a way to discuss what simplify means), for instance, seeing (a+b)/ab as [a/ab + b/ab]. Include context examples as needed - students don’t even remember phone numbers now! (They’re auto programmed.) Be careful with language, “the shortest side is 5” could still allow for TWO sides that are 5. Do you want that as a possibility?
An interesting gender observation that “by and large is true”, from talking to students who have done well on contests: If a male can narrow multiple choice options down to 2 (or even 3) they will take a guess. A female will keep trying to work it out, and if they cannot, they will not guess. There’s a definite taste for problem types, some will engage where others won’t. (Me, I said I might stop once I realize I CAN get to a solution. Got some funny looks for that.)
I finished the morning by chatting with Bruce McLaurin and Jimmy Pai at lunch. One particular item that came up was how many (formerly) good questions can be “googled” now. Ten years ago, you could ask “how many folded sheets does it take to reach the moon”, and not get an answer via a search engine. Now that you can, try to ask question variants that cannot be “googled”, such as “how many folded sheets does it take to reach the roof of the school”.
In conclusion, recreational mathematics still has a place, be it in the classroom or in playing around with math for contests. If not easily incorporated, it could still be a warm up or a bonus, to get students doing (and liking) math. Also, there’s a big link between math mistakes, and how math contests are created. Hope you found this post of use!
I also have a prior COMA conference post featuring Marian Small which you could check out.