Friday, June 15, 2018

2018 Year in Review

Unlike the last three years  (2017,  20162015)  I volunteered at the local Middle School for the first time.  So this year's reflection is all about that transition from working with upper elementary students to actual teenagers.   Going into the process I wondered whether they would be easier or harder to lead. I remember thinging "On one hand, they should be more mature and capable of greater focus on the other hand the teenage years can be stressful."   Likewise, because I planned to run one group for all the grades I was very worried about  how to provide accessible content that would be fun and interesting for everyone.

Overall I ended the year very pleased. One of the most striking observations for me was the great shift in maturity (and height) between the sixth graders and eight graders.   Sixth grade to generalize is really part of elementary school and  eighth grade is basically high school.  The sixth graders in many ways were similar to what I've seen in the younger grades. They were enthusiastic, sometimes boisterous, loved to joke but also needed more compelling activities to stay on task. The older students by contrast required much less supervision and had a lot more focus and drive.  I ended up often relying on the eight graders as a backbone of maturity and leadership for the group as a whole. Next year will be interesting in this respect, since I expect most of the rising sixth graders to continue and to pull in a new group of rising fifth graders but I only have 1 current 7th grader. I'd really like to recruit a larger pool to anchor the group again.  My current idea is to reach out to the 8th grade Math teacher and see if I can have him give out a flier to his class.

Groups and Questions

However, one of the trends that did follow from the heterogeneous grouping was the older kids tended to work together. From time to time I gently encouraged some mixing but I think also this is understandable and a social fact  I need to keep in mind. I have to maintain enough of a balance of ages in the room so that no one grade dominates and the kids feel like there is a progression where they will have grade peers as they go forward. This sort of parallels how I feel about gender. Its important to maintain enough of a balance that both boys and girls feel comfortable in the room. After that is achieved, if I don't intervene the girls will tend to work more often with each other and vice versa. So I nudge from time to time but I'm still mostly wedded to the idea that in a club where everyone comes voluntarily I shouldn't obsessively force interactions and that regular classes fulfill this function.  If everyone is actively engaged I'm happy.

Along these lines where I'm most  strongly considering a shift is in how I do questions. For the same reasons as groupings I don't normally cold call. Instead I try to actively track who has answered and call on other kids if possible. I also pull kids aside before hand and try to encourage them to go in front of the group. But even using these strategies I wasn't completely satisfied with participation in group discussions this year. There were clearly 4-5 kids who didn't want to talk and I think integrating them more fully in is worth some risks.  So I'm considering just announcing that I will call on everyone for group questions and white boarding as part of our club norms next year and explaining why.  I'm also going to dovetail this with some discussion about public mistakes since that is definitely part of this phenomena.  I still have to think about to this works for introverts (I definitely had some this year)


As I mentioned previously, in the beginning of the year I worried a lot about the huge potential gap in Mathematical backgrounds of the kids I would attract. In theory I could pull kids taking Math 6 all the way up to Algebra II. That encompasses a huge 5 year spread of classes. What happened in practice was almost everyone who joined had completed or was taking Algebra I.  This simplified my planning process quite a bit since I didn't worry as deeply about introducing too much Algebra along the way and I tried to tune for the general 3 year spread in classes.  However, I did have one Math 8 student where this didn't work out well. She was very quiet and reluctant to work with the other kids. Despite trying to work with her each week  I didn't realize until half way through that she felt lost at times and wasn't confident enough to ask questions. For me this was partly a reminder to ask more questions myself. But I think also narrowing the focus down will help here. If I change the club charter to be for kids at the Algebra I level or higher or at least comfortable with asking questions if an occasional Algebra concept is introduced I will be able to plan with better precision.   Long term, If I see more demand for a Pre-Algebra focused club I think it  would sense to find another adviser and run a second group. This split is very difficult to bridge otherwise.

Based on specific feedback from the kids I was pleased several of the topics I took risks experimenting with like  the King Chicken problem from (here) or the day on Polynomial Deltas from ( here)  were well received.   That's encouraging because I have a full year of meetings to fill and some kids have been with me now 3 years in a row.  I'm thinking about how to mine more Math Circle topics from various books and also on the lookout for some more art projects to intersperse. Tessellation based images are one of the first to come to mind.  I think the math history day (here) was also successful and I will think more about ways to bring history in from time to time.


I transitioned away from several local contests  I did not really like this year and replaced them with 3 new ones:

  • MathCounts
  • Purple Comet
  • UW Math Hour Olympiad
In general these all were improvements.  For the most part I think I get more bang for the buck doing a contest during a club meeting. That maximizes participation which by itself is worth a lot. But in addition, it allows the potential for reviewing problems together afterwards as a group and de-emphasizes the award/ranking ceremonies.  If the kids win they of course love the trophies but for the most part I think the potential for discouragement remains high. In addition, we can still leverage all the benefits of competition in houses.  Calling something a contest still creates drive and excitement and often brings out the best work in students.  

Despite being traditional in format Math Counts was a great success in particular. The kids mentioned it repeatedly when I asked for feedback.  The question banks are generally superior to the local ones so I'm mostly happy with it. However,  I did have some kids fixate on a rivalry with another school. So framing the competition and how we treat it remains an issue to work on next year.   

Finally, I had one request for a training schedule for the various competitions. In the past I've assumed that the kids really wouldn't have the time or drive to do large amounts of additional practice at home so I haven't done more than send out links to old tests.  Next year I'm going to try an experiment and make a light suggested practice schedule for AMC8 on a spreadsheet and let the kids signoff on what they tried.  We'll see from there if there is more interest that I realized.

This Year's Topic Map

Friday, June 8, 2018

More connections

I've been thinking alot about polynomial deltas recently. See:  It turns out, that there are a variety of problems where its fun to use them. Basically anywhere you think you have a polynomial function and you can curve fit is a good candidate.

For example:  Find a formula for \( \sqrt{n\cdot (n+1) \cdot (n + 2) \cdot (n+3) + 1} \)

You could do the algebra and factor cleverly or you could calculate the easy values around 0,1,2 ... and calculate the deltas to do a quick fit.

But I thought of another scenario this morning where I think they come in particularly nicely and answer a long standing philosophical question of mine. There's a class of formulas that are usually proven inductively where one's often left asking: "How did someone find the original pattern to test?" As a student I would just play around, but now I see these more as curve fitting exercises.

A good example of this is the sum of squares \( \sum_{i=1}^{n}n^2   = \frac{n (n+1)(2n +1)}{6}\)

The inductive proof is not hard, and there are some beautiful visual versions (link to proof ) but it was always hard for me to think how this was actually discovered.   Enter the deltas ....

When looking for a formula we just need to generate enough values and see if the deltas resolve. If they do its a nth degree polynomial and we can work out the coefficients.

n   sum-of-squares  deltas

0    0
1    1            3
             4           2
2    5            5
             9           2
3   14           7
4   30

This shows its a 3rd degree polynomial of the form  \( Ax^3 + Bx^2 + Cx + D\)   

  • from f(0) = 0 we see D = 0 
  • from the deltas we see  \( A = \frac{2}{6!} = \frac{1}{3} \)  
  • We can then substitute in f(1) and f(2) to get a simple system \(B + C = \frac{2}{3} \) and \(4B + 2B = \frac{7}{3} \)
  • After solving we find: \( f(x) = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{6} \)  which combines to exactly our original  \( \frac{n (n+1)(2n +1)}{6}\)

Note: you could also treat this like a linear system if you can tell what degree the function is likely to be but that's actually more work anyway in many cases.

Wednesday, June 6, 2018

6/5 Woven Math

I've been wanting to do this math/art project since I first saw Allison's artwork on twitter. I finally had enough time to practice and find the supplies.  Once you gain experience its possible to make all of the Platonic solids but 1 or perhaps 2 basic shapes are probably a good target for the first time.

  • 2 large poster boards sufficed for 15 kids and I think could have easily made twice as many structures.
  • We spent about 45 minutes and everyone made 1 or 2 projects.
  • Keep repeating over/under at each step and make sure to look for the correct woven pattern before taping.

Preparing strips ahead of time would be great but it was too time consuming so I had the kids bring rulers and scissors and built in time for prep work. We ended up dividing into drafter, cutter and weaver teams. As the strip cutters built up enough supply, I switched them over to weaving.

We used 1/2 inch strips that were 5 inches long.  You could probably go an inch or so longer. The kids liberally secured each joint with tape.



The basic weaving joint.


Add 3 more to make a double cross.   And then 2 more to make the first triangle.

2 more strips to reach the second triangle and 1 more to make the half ball. Finally join  the tricky last 4 strips.

Friday, June 1, 2018

5/29 Phi Day

This week I wanted to extend some of our talk about the golden ratio. For the last reference see:   I'm also not quite done testing out the weaving activity so this was easier to schedule right after a busy Memorial Day weekend.

I had two videos in mind that I recently saw:

I also had several group demos that I wanted to start off with. But I started by asking the room what they already knew about the golden ratio. As expected, Fibonacci numbers were mentioned and facts about famous art examples i.e. the Mona Lisa.

That was a good bridge to start with a precise definition of the ratio and from there we covered:
  • Phi is not the same as the Fibonacci numbers and in fact for all such sequences defined by \( F_n = F_{n-1} + F_{n-2} \) the ratio \( \frac{F_n}{F_{n-1}}\)  tend to approach the golden ratio.  This one you can test with your own generator numbers.  
  • Phi in the Pentagon. We derived the basic ratio  of the diagonal (upper center diagram) as a group.
Image result for pentagon and golden ratio

  • The general properties of Phi based on its root equation \(x^2 -x -1 = 0 \) i.e.
$$ \Phi^2 = \Phi + 1 $$
$$  \Phi =  1 + \frac{1}{\Phi} $$

  • The idea of finding spirals in a square.   I gave some demo with finding facts about the 10x10 grid. For example divide diagonally and you get its the sum of 2 triangle numbers. The challenge I gave out was for the kids to find a way to break it into a spiral.  Interestingly we found both ways of counting 10 + 9 + 9 + 8 + 8 vs. 10  +  8 + 10 .... which led to an interesting discussion ("You're both right - how can that be?")
I worked these on the board with breaks for the kids to try things out at their tables.  We then watched the videos.  Overall while everyone watched attentively, the first spiral simulation was particularly appreciated,  I think doing both was a mistake and I should have only used the numberphile one. On reflection, there was  a little too much passive viewing  and I would build a final Phi investigation/activity in at the end instead if I repeat.  The general idea of breaking a rectangle into its spiral/continued fraction is probably enough for an entire day on its own.

POTW adapted from Mike Lawler

1.  Can you find a polynomial with all integer coefficients and one root equal to ?
2. Can you find a polynomial with all integer coefficients and one root equal to ?
3.  Can you find a polynomial with all integer coefficients and one root equal to ?

[Notes: this was a bit too involved for a take home problem or perhaps its just the end of the year setting in. Next time it could probably be combined with the polynomial deltas to make a complete polynomial day.   In practice, I ended up carefully working part 1 as a group the next week and talked alot about conjugates and the relationship between roots and factors.]

Wednesday, May 30, 2018

5/22 Cycles and Circles

[Memorial Day delayed me getting this one out. Hopefully it was worth the wait.]

Today's theme was circle and cycles. The main motivation was the "King Chicken" graph theory problem which I'll describe below. But after brainstorming a few other semi-related ideas came to mind that I thought would make a coherent session. I also experimented a little bit with format this time. I really wanted a "station" where I could work one on one for a bit longer than normal with kids. So I decided to setup the room with whiteboard problems and have the kids move among the problems and the table where I was curating the graph theory problem. This worked fairly well. I was able to focus more on the problem I wanted to highlight in a small group. The flip side was I did have to get up and refocus a few kids more often than I would have circulating around and I had less insight into group thinking on this part beyond the whiteboard artifacts (But these were all fairly interesting).

VNPS  Carnival

I worked through the beginning of the first problem  as a group to get everyone going.

Divisibility by 7 Non Planar
[Tanya Khovanova]

The 7 Divisibility Graph: To find the remainder on dividing a number by 7, start at node 0, for each digit D of the number, move along D black arrows (for digit 0 do not move at all), and as you pass from one digit to the next, move along a single white arrow.

After trying this out with the kids supplying some test numbers. I asked them to consider why it worked and if they could come up with a similar graph for divisibility by 13.

Next to this was a geometry problem from "Geometry Snacks" This was probably my weakest thematic link but provided a needed problem and some more variety.

The outer circle is unit circle. There are 4 medium circles B,C,D, and E and 1 small inner one A. All the circles are tangent with each other as shown. What is the smallest circle radius?

Next came a return to the cyclic / graph space with a problem that I suspected was not new for some of the kids. So I added a part 2) with a less well known extension. This one generated the rather interesting circle art in the original photos from one student who was connecting evenly spaced participants on a circle to each other.

Part 1.

N people in a room each shake hands with each other - how many total?

Part 2.

Show that there will always be two people at the party, who have shaken hands the same number of times.

Chicken pecking probability

This was a great linkage and chosen for its connections to "King Chicken".

The question: In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks?

King Chicken

See the middle of:

The main idea is we define a strict pecking order between chickens in a coop and then explore the graph using the idea of "King Chicken" as a motivator.

I tended to emphasize coming up with a definition of what a King Chicken is first. Most kids arrived at the idea of pecking the most other chickens. The thing to emphasize is I'd like a definition where it defines a relationship between a chicken and every other one not just most of the other ones. From there I had the kids explore sample graphs on a size 5 flock:

Most of the time was spent on developing ideas about whether we could find configurations with all the combinations 1 King, 2 Kings through 5 Kings.  This by itself was probably a 30 minute exercise and engrossing.


This geometry puzzle is actually a bit harder than I realized:

Tuesday, May 29, 2018

Exponent Tower Redux

I saw a different version of the tower of 7's problem in a book I'm reading on number theory. This is the mostly rambling thought process I've been going through:

Quick Review: What are the last 2 digits of  \( 7^{7^7} \) ?

My quick and dirty original solution:

  1. Make a table of the multiples of 7 and notice they cycle every 4 terms.
  2. So the problem reduces to figuring out the linear congruence \( x \equiv 7^7 \mod 4 \)
  3. \( 7^7  \equiv 3^7 \equiv (-1)^7 \equiv -1 \equiv 3  \mod 4 \)
  4.  So the answer is the last 2 digits of 7^3 or 43.
Comparison Technique:

1.  First compute the totient function for 100: \( \phi(100) = \phi(2^2)\phi(5^2) = (2^2 - 2^1)(5^2 - 5^1) = 40 \).
2. So everything cycles in 40.
3. Therefore if \( x \equiv 7^7 \mod 40 \) then  \( 7^{7^7} \equiv 7^x \mod 100 \) 
4. \(7^7 = 7^4 \cdot 7^2 \cdot 7 \equiv 1 \cdot 9 \cdot 7 = 63 \equiv 23 \mod 40 \)
5. Then compute \( 7^{23} \mod 100 \) using squaring of sevens i.e.  a table.

That didn't seem like an improvement to me but it got me thinking about 2 things:

  • Does the totient function just bound the cycle length of  relatively prime numbers? 
  • What's the relationship between it and the true cycle lengths?

Let's just look at mod 100 i.e. the last two digits of a number raised to various powers:

Some data:

Seed: 3 CycleLen: 20 CoSeed:97 CoCycleLen:20
Seed: 7 CycleLen: 4 CoSeed:93 CoCycleLen:4
Seed: 9 CycleLen: 10 CoSeed:91 CoCycleLen:10
Seed: 11 CycleLen: 10 CoSeed:89 CoCycleLen:10
Seed: 13 CycleLen: 20 CoSeed:87 CoCycleLen:20
Seed: 17 CycleLen: 20 CoSeed:83 CoCycleLen:20
Seed: 19 CycleLen: 10 CoSeed:81 CoCycleLen:5
Seed: 21 CycleLen: 5 CoSeed:79 CoCycleLen:10
Seed: 23 CycleLen: 20 CoSeed:77 CoCycleLen:20
Seed: 27 CycleLen: 20 CoSeed:73 CoCycleLen:20
Seed: 29 CycleLen: 10 CoSeed:71 CoCycleLen:10
Seed: 31 CycleLen: 10 CoSeed:69 CoCycleLen:10
Seed: 33 CycleLen: 20 CoSeed:67 CoCycleLen:20
Seed: 37 CycleLen: 20 CoSeed:63 CoCycleLen:20
Seed: 39 CycleLen: 10 CoSeed:61 CoCycleLen:5
Seed: 41 CycleLen: 5 CoSeed:59 CoCycleLen:10
Seed: 43 CycleLen: 4 CoSeed:57 CoCycleLen:4
Seed: 47 CycleLen: 20 CoSeed:53 CoCycleLen:20
Seed: 49 CycleLen: 2 CoSeed:51 CoCycleLen:2

You can see here despite the totient being 40 the actual max cycle length is 20 and by necessity all possible cycle lengths are factors of 40.

The mostly symmetric relation between a number n and its counterpart 100 - n is also apparent.
This follows from \( (p-n)^2 = p^2 - 2np + n^2 \equiv n^2 \mod 100 \) Which only directly affects the first term but the effect is noticeable.

Which has led me to: which depends on LCM of the two sub-totient functions and gives:

\( \lambda(100) = 20 \)

Its also interesting what happens as the modulus is varied:

7 is not nearly so "quick"

Mod 99

Seed: 2 CycleLen: 30 CoSeed:97 CoCycleLen:15
Seed: 4 CycleLen: 15 CoSeed:95 CoCycleLen:30
Seed: 5 CycleLen: 30 CoSeed:94 CoCycleLen:30
Seed: 7 CycleLen: 30 CoSeed:92 CoCycleLen:30
Seed: 8 CycleLen: 10 CoSeed:91 CoCycleLen:5
Seed: 10 CycleLen: 2 CoSeed:89 CoCycleLen:2
Seed: 13 CycleLen: 30 CoSeed:86 CoCycleLen:30
Seed: 14 CycleLen: 30 CoSeed:85 CoCycleLen:30
Seed: 16 CycleLen: 15 CoSeed:83 CoCycleLen:30
Seed: 17 CycleLen: 10 CoSeed:82 CoCycleLen:5
Seed: 19 CycleLen: 10 CoSeed:80 CoCycleLen:10
Seed: 20 CycleLen: 30 CoSeed:79 CoCycleLen:30
Seed: 23 CycleLen: 6 CoSeed:76 CoCycleLen:6
Seed: 25 CycleLen: 15 CoSeed:74 CoCycleLen:30
Seed: 26 CycleLen: 10 CoSeed:73 CoCycleLen:10
Seed: 28 CycleLen: 10 CoSeed:71 CoCycleLen:10
Seed: 29 CycleLen: 30 CoSeed:70 CoCycleLen:15
Seed: 31 CycleLen: 15 CoSeed:68 CoCycleLen:30
Seed: 32 CycleLen: 6 CoSeed:67 CoCycleLen:3
Seed: 34 CycleLen: 3 CoSeed:65 CoCycleLen:6
Seed: 35 CycleLen: 10 CoSeed:64 CoCycleLen:5
Seed: 37 CycleLen: 5 CoSeed:62 CoCycleLen:10
Seed: 38 CycleLen: 30 CoSeed:61 CoCycleLen:30
Seed: 40 CycleLen: 30 CoSeed:59 CoCycleLen:30
Seed: 41 CycleLen: 30 CoSeed:58 CoCycleLen:15
Seed: 43 CycleLen: 6 CoSeed:56 CoCycleLen:6
Seed: 46 CycleLen: 10 CoSeed:53 CoCycleLen:10
Seed: 47 CycleLen: 30 CoSeed:52 CoCycleLen:30
Seed: 49 CycleLen: 15 CoSeed:50 CoCycleLen:30
Seed: 50 CycleLen: 30 CoSeed:49 CoCycleLen:15
Seed: 52 CycleLen: 30 CoSeed:47 CoCycleLen:30
Seed: 53 CycleLen: 10 CoSeed:46 CoCycleLen:10
Seed: 56 CycleLen: 6 CoSeed:43 CoCycleLen:6
Seed: 58 CycleLen: 15 CoSeed:41 CoCycleLen:30
Seed: 59 CycleLen: 30 CoSeed:40 CoCycleLen:30
Seed: 61 CycleLen: 30 CoSeed:38 CoCycleLen:30
Seed: 62 CycleLen: 10 CoSeed:37 CoCycleLen:5
Seed: 64 CycleLen: 5 CoSeed:35 CoCycleLen:10
Seed: 65 CycleLen: 6 CoSeed:34 CoCycleLen:3
Seed: 67 CycleLen: 3 CoSeed:32 CoCycleLen:6
Seed: 68 CycleLen: 30 CoSeed:31 CoCycleLen:15
Seed: 70 CycleLen: 15 CoSeed:29 CoCycleLen:30
Seed: 71 CycleLen: 10 CoSeed:28 CoCycleLen:10
Seed: 73 CycleLen: 10 CoSeed:26 CoCycleLen:10
Seed: 74 CycleLen: 30 CoSeed:25 CoCycleLen:15
Seed: 76 CycleLen: 6 CoSeed:23 CoCycleLen:6
Seed: 79 CycleLen: 30 CoSeed:20 CoCycleLen:30
Seed: 80 CycleLen: 10 CoSeed:19 CoCycleLen:10
Seed: 82 CycleLen: 5 CoSeed:17 CoCycleLen:10
Seed: 83 CycleLen: 30 CoSeed:16 CoCycleLen:15
Seed: 85 CycleLen: 30 CoSeed:14 CoCycleLen:30
Seed: 86 CycleLen: 30 CoSeed:13 CoCycleLen:30
Seed: 89 CycleLen: 2 CoSeed:10 CoCycleLen:2
Seed: 91 CycleLen: 5 CoSeed:8 CoCycleLen:10
Seed: 92 CycleLen: 30 CoSeed:7 CoCycleLen:30
Seed: 94 CycleLen: 30 CoSeed:5 CoCycleLen:30
Seed: 95 CycleLen: 30 CoSeed:4 CoCycleLen:15
Seed: 97 CycleLen: 15 CoSeed:2 CoCycleLen:30
Seed: 98 CycleLen: 2 CoSeed:1 CoCycleLen:1

Mod 101

Seed: 2 CycleLen: 100 CoSeed:99 CoCycleLen:100
Seed: 3 CycleLen: 100 CoSeed:98 CoCycleLen:100
Seed: 4 CycleLen: 50 CoSeed:97 CoCycleLen:25
Seed: 5 CycleLen: 25 CoSeed:96 CoCycleLen:50
Seed: 6 CycleLen: 10 CoSeed:95 CoCycleLen:5
Seed: 7 CycleLen: 100 CoSeed:94 CoCycleLen:100
Seed: 8 CycleLen: 100 CoSeed:93 CoCycleLen:100
Seed: 9 CycleLen: 50 CoSeed:92 CoCycleLen:25
Seed: 10 CycleLen: 4 CoSeed:91 CoCycleLen:4
Seed: 11 CycleLen: 100 CoSeed:90 CoCycleLen:100
Seed: 12 CycleLen: 100 CoSeed:89 CoCycleLen:100
Seed: 13 CycleLen: 50 CoSeed:88 CoCycleLen:25
Seed: 14 CycleLen: 10 CoSeed:87 CoCycleLen:5
Seed: 15 CycleLen: 100 CoSeed:86 CoCycleLen:100
Seed: 16 CycleLen: 25 CoSeed:85 CoCycleLen:50
Seed: 17 CycleLen: 10 CoSeed:84 CoCycleLen:5
Seed: 18 CycleLen: 100 CoSeed:83 CoCycleLen:100
Seed: 19 CycleLen: 25 CoSeed:82 CoCycleLen:50
Seed: 20 CycleLen: 50 CoSeed:81 CoCycleLen:25
Seed: 21 CycleLen: 50 CoSeed:80 CoCycleLen:25
Seed: 22 CycleLen: 50 CoSeed:79 CoCycleLen:25
Seed: 23 CycleLen: 50 CoSeed:78 CoCycleLen:25
Seed: 24 CycleLen: 25 CoSeed:77 CoCycleLen:50
Seed: 25 CycleLen: 25 CoSeed:76 CoCycleLen:50
Seed: 26 CycleLen: 100 CoSeed:75 CoCycleLen:100
Seed: 27 CycleLen: 100 CoSeed:74 CoCycleLen:100
Seed: 28 CycleLen: 100 CoSeed:73 CoCycleLen:100
Seed: 29 CycleLen: 100 CoSeed:72 CoCycleLen:100
Seed: 30 CycleLen: 50 CoSeed:71 CoCycleLen:25
Seed: 31 CycleLen: 25 CoSeed:70 CoCycleLen:50
Seed: 32 CycleLen: 20 CoSeed:69 CoCycleLen:20
Seed: 33 CycleLen: 50 CoSeed:68 CoCycleLen:25
Seed: 34 CycleLen: 100 CoSeed:67 CoCycleLen:100
Seed: 35 CycleLen: 100 CoSeed:66 CoCycleLen:100
Seed: 36 CycleLen: 5 CoSeed:65 CoCycleLen:10
Seed: 37 CycleLen: 25 CoSeed:64 CoCycleLen:50
Seed: 38 CycleLen: 100 CoSeed:63 CoCycleLen:100
Seed: 39 CycleLen: 20 CoSeed:62 CoCycleLen:20
Seed: 40 CycleLen: 100 CoSeed:61 CoCycleLen:100
Seed: 41 CycleLen: 20 CoSeed:60 CoCycleLen:20
Seed: 42 CycleLen: 100 CoSeed:59 CoCycleLen:100
Seed: 43 CycleLen: 50 CoSeed:58 CoCycleLen:25
Seed: 44 CycleLen: 20 CoSeed:57 CoCycleLen:20
Seed: 45 CycleLen: 50 CoSeed:56 CoCycleLen:25
Seed: 46 CycleLen: 100 CoSeed:55 CoCycleLen:100
Seed: 47 CycleLen: 50 CoSeed:54 CoCycleLen:25
Seed: 48 CycleLen: 100 CoSeed:53 CoCycleLen:100
Seed: 49 CycleLen: 50 CoSeed:52 CoCycleLen:25
Seed: 50 CycleLen: 100 CoSeed:51 CoCycleLen:100
Seed: 51 CycleLen: 100 CoSeed:50 CoCycleLen:100
Seed: 52 CycleLen: 25 CoSeed:49 CoCycleLen:50
Seed: 53 CycleLen: 100 CoSeed:48 CoCycleLen:100
Seed: 54 CycleLen: 25 CoSeed:47 CoCycleLen:50
Seed: 55 CycleLen: 100 CoSeed:46 CoCycleLen:100
Seed: 56 CycleLen: 25 CoSeed:45 CoCycleLen:50
Seed: 57 CycleLen: 20 CoSeed:44 CoCycleLen:20
Seed: 58 CycleLen: 25 CoSeed:43 CoCycleLen:50
Seed: 59 CycleLen: 100 CoSeed:42 CoCycleLen:100
Seed: 60 CycleLen: 20 CoSeed:41 CoCycleLen:20
Seed: 61 CycleLen: 100 CoSeed:40 CoCycleLen:100
Seed: 62 CycleLen: 20 CoSeed:39 CoCycleLen:20
Seed: 63 CycleLen: 100 CoSeed:38 CoCycleLen:100
Seed: 64 CycleLen: 50 CoSeed:37 CoCycleLen:25
Seed: 65 CycleLen: 10 CoSeed:36 CoCycleLen:5
Seed: 66 CycleLen: 100 CoSeed:35 CoCycleLen:100
Seed: 67 CycleLen: 100 CoSeed:34 CoCycleLen:100
Seed: 68 CycleLen: 25 CoSeed:33 CoCycleLen:50
Seed: 69 CycleLen: 20 CoSeed:32 CoCycleLen:20
Seed: 70 CycleLen: 50 CoSeed:31 CoCycleLen:25
Seed: 71 CycleLen: 25 CoSeed:30 CoCycleLen:50
Seed: 72 CycleLen: 100 CoSeed:29 CoCycleLen:100
Seed: 73 CycleLen: 100 CoSeed:28 CoCycleLen:100
Seed: 74 CycleLen: 100 CoSeed:27 CoCycleLen:100
Seed: 75 CycleLen: 100 CoSeed:26 CoCycleLen:100
Seed: 76 CycleLen: 50 CoSeed:25 CoCycleLen:25
Seed: 77 CycleLen: 50 CoSeed:24 CoCycleLen:25
Seed: 78 CycleLen: 25 CoSeed:23 CoCycleLen:50
Seed: 79 CycleLen: 25 CoSeed:22 CoCycleLen:50
Seed: 80 CycleLen: 25 CoSeed:21 CoCycleLen:50
Seed: 81 CycleLen: 25 CoSeed:20 CoCycleLen:50
Seed: 82 CycleLen: 50 CoSeed:19 CoCycleLen:25
Seed: 83 CycleLen: 100 CoSeed:18 CoCycleLen:100
Seed: 84 CycleLen: 5 CoSeed:17 CoCycleLen:10
Seed: 85 CycleLen: 50 CoSeed:16 CoCycleLen:25
Seed: 86 CycleLen: 100 CoSeed:15 CoCycleLen:100
Seed: 87 CycleLen: 5 CoSeed:14 CoCycleLen:10
Seed: 88 CycleLen: 25 CoSeed:13 CoCycleLen:50
Seed: 89 CycleLen: 100 CoSeed:12 CoCycleLen:100
Seed: 90 CycleLen: 100 CoSeed:11 CoCycleLen:100
Seed: 91 CycleLen: 4 CoSeed:10 CoCycleLen:4
Seed: 92 CycleLen: 25 CoSeed:9 CoCycleLen:50
Seed: 93 CycleLen: 100 CoSeed:8 CoCycleLen:100
Seed: 94 CycleLen: 100 CoSeed:7 CoCycleLen:100
Seed: 95 CycleLen: 5 CoSeed:6 CoCycleLen:10
Seed: 96 CycleLen: 50 CoSeed:5 CoCycleLen:25
Seed: 97 CycleLen: 25 CoSeed:4 CoCycleLen:50
Seed: 98 CycleLen: 100 CoSeed:3 CoCycleLen:100
Seed: 99 CycleLen: 100 CoSeed:2 CoCycleLen:100
Seed: 100 CycleLen: 2 CoSeed:1 CoCycleLen:1

Another digression:
The odd numbers (often/always) cycle through each other for the last digit when raised to a power mod 10^n and if relatively prime we need to get to 1 before the cycle restarts so once you reach a number ending in one you can just start successively squaring it.   [Is this generally true? [no]]

And that's about where I'm at currently.

Friday, May 25, 2018

I'm going to a Conference!

Dear Benjamin Leis,

On behalf of the 2018 Northwest Mathematics Conference Program Committee, we are pleased to inform you that your session, “Middle School and Math Circles”, has been accepted! An email will be sent with your specific presentation date & time by mid-June. Edits to your title and/or description may be made by the Program Committee.

We received an overwhelming number of proposals — more than twice as many proposals as we were able to accommodate. Please confirm your acceptance by June 30, 2018. If you have not confirmed by this date, we will begin to accept speakers from our lengthy waitlist.

As a lead speaker, your registration is complimentary and will be completed for you upon your confirmation of acceptance. If you have a co-speaker/s, they will need to register through the conference website, unless they are a lead speaker for a different session. The breakfast keynote on Saturday morning with Annie Fetter is not included in your complimentary speaker registration. If you indicate on the confirmation page that you are interested in attending this keynote, information will be emailed to you at a later date.

Thank-you for your willingness to share your ideas and contribute to the richness of the program for the 2018 Northwest Mathematics Conference hosted by the BCAMT.

Chris Hunter & Janice Novakowski
Program Committee
2018 Northwest Mathematics Conference

Sunday, May 20, 2018

Tangles and Symmetry

This is a description of Dr. David Pengelley's talk "All Tangled up and Searching for the Beauty of Symmetry" which I just attended. This makes an excellent Math Circle topic and I think I might use it next year.


1. 3 different colors of dental floss or some equivalent string cut into 3 foot lengths.
2. A 6-8 inch wooden dowel
3. A note card with 3 holes for the strings and with an X on one side and an O on the other.

Its helpful to tape the string in bundles of 3 to keep them from tangling before handing them out.

Notecard Explorations

1. First hand out the note cards. Have the kids explore how many different states of the note card there are with same orientation.

There are 4 (2 with the x showing on top and bottom and 2 with o showing on top and bottom.) Note: the rectangle must stay with the longer side vertical.

2. Next explore how many ways there are to transition between states.

  • Flip, Rotate and Spin and most importantly None. 

Make a an operation chart to explore the transitions:

Things to look for: closure, identity element, commutativity. i.e. this is an Abelian Group.

Full Space (Quarternions)

3. Next add the strings. Each string should be threaded through the prepunched holes and tied to the dowel.

4. Next explore whether anything has changed. Key question are the strings the same after a flip, rotate etc?

Introduce notion of clockwise/counterclockwise transitions. Also Full rotation = 2 in one direction.

5. Can we get back to the original state w.r.t to the strings and card after 2 and 4 rotations? You are allowed to move the strings but not the dowel or card and the strings may only move around the card. Split group in half and have each piece work on part of the problem. Remember to pre-practice play with the transitions before hand so they are very familiar.

(Only 4 rotations works. Several ways to move the strings to prove it. The easiest is to take all of them and move them around the card.) This takes some time.

6. What state does a flip / rotate end up in. (Either a forward or backwards flip.

7. Build the new operations table:

8 states: None, Full Turn, Rotate Clockwise, Rotate Counter Clockwise, Forward Flip, Backwards Flip, Clockwise Spin, Counter Clockwise Spin.

Note new patterns: This is not commutative for example. How do both tables relate to addition and subtraction?

Wrap up: Tie to Algebraic Group and Quarternion I like bringing in a bit of math history about Hamilton and the Broome Bridge. See:

Wednesday, May 16, 2018

5/15 Chaos + Mod Arithmetic

We started the day looking at the problem of the week (from @mpershan):
Given a triangle with side length A, B, and C 
  • If A/B = B/C = 1, then it's an equilateral triangle. 
  • If A/B = B/C = 2, it can't be a triangle. 
What the largest value of A/B = B/C it's possible for a triangle to have?

I tried a slightly different structure this time and instead of asking for solution demos I asked the room for what they noticed about the problem.  This was actually fairly productive.   We started with several statements about the triangle inequality. That wasn't generally known so I demoed it on the board.  I like creating pictures where the two smaller sides are very short so its really clear they can't meet.  We then had the idea presented to fix one of the sides to length 1 and see what happens. Eventually the kids experimented with concrete numbers for the ratio of two and found the results: 1,2,4 violated the triangle inequality. Finally, one student came up with the general inequality x^2 < x + 1.  I had another one solve it via the quadratic formula and to cap everything off I asked if anyone recognized the result.  I ended with a small speech on the golden ratio and how it shows over and over again in unexpected places. [I may go down this path for a future theme.]

Based on an interesting post by Matt Enlow: I inserted a group activity next:

In an nutshell:

  • Start with an equilateral triangle.
  • Pick a random point within it.
  • Randomly select a vertex and find the next point half way between the  last one and that vertex.
  • Repeat

After thinking about  some previous feedback comments I chose to do the simulation on a very large communal piece of butcher block paper  (3 ft x 3ft).  If I had a larger sheet I would have gone even bigger.  I drew an equilateral triangle using a standard intersection of 2 arcs methods with a tape measure which actually elicited some discussion. Then everyone crowded around the side of the paper and we took turns rolling the dice, measuring the next point and marking it. (This also let me keep a rough tab that the points were  accurate.)  At strategic points I had the kids make predictions about what they observed.   We generated over 30 points which takes a while but gave everyone a chance to roll the dice at least twice.  This is enough to see the beginning but not the full pattern.  But the kids were already able to make a good conjecture about why you couldn't get back to the center. After this point I switched to an internet simulation:

This let me run thousands of random choices and show the emerging Sierpinski triangle. As I hoped this produced a lot of spontaneous "Wows"  We had a bit more followup fractal discussion but if I repeated I would love to find a second half that thematically linked here.

Finally I returned to my original plan to go over a bit of Modular arithmetic using a Math Circle structure from    This is a bit dry for Middle School although I like the starting magic trick and that did work well. What  I ended up doing was printing enough packets for pairs to work on and doing the first few questions as a group with some  structured lecturing on my part and then circulating to help groups work through the later parts of the set. Overall: I kept the room moving and I finally introduced the topic which was a goal but I think there is more room to grow this session in the future.    I think focusing on multiplication and addition tables is one of the more natural pieces to use and perhaps dropping some of the formal linear congruence proofs. For almost all of these, I found I was suggesting the kids try concrete numbers first and observe patterns to get at what the packet was suggesting.

Thursday, May 10, 2018

5/9 Square Roots

Today I returned to a topic, square roots, I've done before with younger kids :  My thinking was that I had a cool video I wanted to show that mentioned approximating the numeric value of a square root and doing it ourselves would motivate that part of the video and provide some embedded practice calculating with decimals.

But before we could start in we needed to go over the old problem of the week.

"In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three 5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile. Find the number of different positive weights of chemicals that Gerome could measure"

This one lent itself really well to whiteboard demos and I had kids present two different approaches. One was combinatoric, the other just brute force listed the cases. Since there is a max of 168 total in weight and only 9 weights either way is very approachable. However, none of the kids thought about putting the weights on both sides which presented me with a dilemma. I decided to approach it as follows "All the ideas you've mentioned are correct with some assumptions, lets make those explicit and double check if we can relax any of them." That way I could honor the thinking already done and point out the further avenue to explore.

Next we started on the square root investigation. I framed the problem as follows:

"How does the square root key work on the calculator? What do you think its doing? I'm going to put up some simple square roots \(\sqrt{2}, \sqrt{5}, \sqrt{7}\) can you find an approach to calculate value of these to a few decimal places (without a calculator)?"

What followed was a fairly useful exercise on two levels. There was a ton of calculation practice and several variants on the bound and search for a better fit algorithm emerged. At the end I also raised the question "For what other calculator keys do you wonder about how they work and are there any themes in the implementations?"  Logarithms were probably the best candidate for a future session. One possibility is stressing the use of series and iterative algorithms.

From there I tried to fit in an old problem on the same theme I've been saving:

Find the cube root of \(x^6  - 9x^5 + 33x^4 - 63x^3 + 66x^2 - 36x + 8 \).   This we ended up doing as a group discussion. The kids eventually found the first and last coefficients of the root Ax^2 + Bx + C  but were stuck on the middle one B.  I really wanted to carefully go through the distributive law work and finish the solution but I was short on time and had to call this to a close. Moral: This is more than a 10 minute problem (and maybe more problems requiring polynomial mult. are called for)

In the midst of all of this I had audio issues with the projector and had to rush around to find another room we could use with a working system. So we filed over next door and spent the last 25 minutes watching this really fascinating 3Blue1Brown video which I alluded to originally:

Right at the beginning the square root approximation problem is discussed and hopefully it had extra resonance after trying it ourselves.


A fun triangle inequality / golden ratio problem from @mpershan:

I've now seen an excellent numberphile video on Phi that I could easily integrate with this. So maybe we'll do a golden ratio day. Although I also want to fit in a 3-D weaving art project as well in the next few weeks.

Friday, May 4, 2018

5/2 Following my passion

This week I ended up switching my focus on the fly. I had been originally been planning on doing a graph theory math circle  activity centered around chicken pecking hierarchies. (Yes really)  But I became so excited about thinking about polynomial deltas that I ended up asking myself the question "Why not go with what you're excited about?" All the kids have enough background with polynomials so we could just jump in which was an added bonus.

I stuck with one part of my planning process. I had already decided I wanted to start with some group white boarding focusing on some of the geometry puzzles from @solvemymaths.

So I picked 3 of them included the now infamous pink triangle.  In each the goal is to figure out which fraction of the shape is shaded pink and usually any polygons are regular.

I put each of these up on different sections of the whiteboards and let the kids circulate among them forming organic groups. (Occasionally I'll nudge kids to work together) They then spent about 15-20 minutes attempting to find solutions while I circulated and interacted with individual groupings.  My particular focus this time was to emphasize thinking about the problems and coming up with ideas.  I used the "What do you notice/wonder?" prompt quite a bit.  There was a lot of good thinking but I definitely still see room for encouraging more experimentation.  At the end of the process I had everyone regroup for a discussion of what various people had found. Interestingly, the first pink triangle solution was analytic i.e. the student setup to equations for the lines and found the intersection.  I think this reflects the emphasis the curriculum places on these type approaches over pure synthetic reasoning.  Students don't see similar triangles quite as quickly. As an aside I had to explain the expression "broke the internet" to one of the boys as in "this puzzle just broke the internet this week."

For the second half, I switched over to looking at the patterns within polynomial deltas. See:  for my motivation.


x^2 + x  + 1

1    3
2    7         2
            6        0
3    13       2
4    21

The way I structured this section was to demonstrate calculating deltas on a sample polynomial and have the kids then come with up with their own polynomials and look for patterns on what was occurring.

  1. How many levels of deltas before you hit you and why?
  2. Is there some pattern to what the second to  last value was etc?
  3. What does this mean? I had one kid talk about velocity/acceleration and I think I would draw this out more if I repeated as well.

When they started coming up with enough ideas, I then suggesting trying to investigate general classes of polynomial like Ax^2 + Bx + C.  This generated lots of good distributive law practice on the whiteboards.

Finally for my 3rd prompt I asked can you go backwards as well as forwards and why do you think it does or doesn't work?

I also put up the original problem from the last post as an extension at the end for a few kids who worked quickly enough to get there.

This structure worked pretty well the one element I would improve on was to add in graphing. I only have a single computer to use but it occurred to me after the fact that I could have done some group desmos activities in the beginning with various polynomials and graphically shown the deltas.

[Another interesting angle to pursue would be function inverses. What's the behavior of the deltas when a function has an inverse vs. when it doesn't can we make a test?]

To close, I mentioned this kind of investigation is closely related to another branch of mathematics and had everyone take guesses at what it was.  As I expected no one even came close to saying Calculus which shows how mysterious it is . 

Tuesday, May 1, 2018

Polynomial Deltas

Find the polynomial  given:

f(1) = 1          f(4) = 3
f(2) = 1          f(5) = 5
f(3) = 2          f(6) = 8

In the past I would have viewed this as a 6 variable linear system and given the complexity gone to linear algebra and matrices probably with a computer assist. 

But I saw a new way to approach this via @eylem_99 that relies on analyzing the deltas.  The idea comes from a tool I've only ever seen used for making sense of polynomials.  Calculate the deltas between values of a func for instance f(2)-f(1) = 0 and then calculate the second level deltas etc. Eventually after n iterations where n is the same as the degree of the equation you'll reach a constant value.  You can prove this via algebra but essentially these deltas are approximations of the various derivatives. When you reach the nth level the derivative becomes constant and the approximation is completely accurate.

So its a fun exercise to do this on various polynomials and see what falls out. What had never occurred to me was this also can be used to reverse additional values for the function by generating the deltas in reverse!

So for example from the above values we can regenerate  f(0), f(-1), and f(-2) starting with constant delta -3 on the right:

-2   110             
-1   36    46    
         -28  -25
0     8    21 11  
          -7 -14         -3
1     1     7 8  
            0 -6          -3
2     1     1 5
            1 -1          -3
3     2     0 2
            1 1          -3
4     3     1 -1
            2 0
5     5     1
6     8
The final delta is actually and note that in general this would 5! * A so
f(0) = 8 so F = 8

Now adding and subtracting complements f(1) and f(-1)  and f(2) and f(-2) generates simpler
linear equations for finding B,C,D, and E.

2B + 2D + 16 = 37                         and -1/20 + 2C + 2E = -35              
32B + 16D + 16 = 111                              -32/20 + 16C + 4E = -109

From there its not to difficult to get the rest of the constants and certainly simpler than the matrix math.

Wednesday, April 25, 2018

4/24 Tic Tac Toe (Naughts and Crosses)

Today started with some administrative tasks. 

  • I had the kids who went to the Quiz bowl talk about their experience and whether it was worth repeating. The consensus is that its fun (although  not particularly math related) So I'd like to figure out a way to do this more next year without sacrificing any club time. The best option would be to spin off another ASB club but that requires another sponsor and quite a bit of leg work.
  • I also asked about the Purple Comet Meet. Again the kids also really enjoyed it so we will do it again next year.
  • A quick review of last week's Problem of the Week.

I had two main goals for the day day: go over a few problems from the Purple Comet meet so kids could see solutions to problems they hadn't figured out or alternatively have a chance to present a solution they found to the group.   I figured that 3 questions would be about the right number from a focus perspective.  

I projected the problem set on the board and the kids chose numbers 4,9 and 20 to look at as a group. The best discussion was on the first problem where we had 3 different solutions presented. On the last I really wanted to draft a usually shy kid to draw the diagram on the board but he was very adamant about not wanting to do so. I didn't insist and had another kid do it instead. I'm going to think more about the best way to do this is. Is it enough to have kids participate in groups with each other or is it important to strongly encourage them to also talk in front of everyone?  As a first step I intend to privately chat and see if I can get him to volunteer in future weeks.

Then for contrast I wanted some game or puzzle to do for the remainder of the time which I assumed we would be around 20-30 minutes.    My problem was I didn't have any solid ideas in mind and on a lark I mentioned that on twitter. In a very affirming moment I received a bunch of interesting suggestions:

A theme emerged around variants of tic-tac-toe that I thought would work very well.

I ended up with 3 versions to have the kids try:

  1. Tic-Tac-Toe on a torus
  2. Ben Orlin's  ultimate Tic Tac Toe.   which is  done with a grid of 3x3 tic tac toe games. This was probably the favorite version for the room.

3. David Butler's 4D Tic-Tac-Toe I had mostly run out of time so we just barely described the rules which are a bit complex.

Overall these were a big hit.  Middle schoolers still find the game fun and I had a few kids I literally had to shoo out of the room past our closing time.

Problem of the Week:
I'm still in a purple comet meet mood so I took a mid-level problem from a previous year that I liked:

Friday, April 20, 2018

4/16 Purple Comet Online Contest

Three years ago I discovered the Purple Comet contest @  It has close links to the AwesomeMath  and I really liked the problems in the old tests. So I tried it out with my fifth graders at the time.  That 2015 Experience discouraged me from doing it again.  Despite the kids theoretically having up to Math7 knowledge the contest was too hard and I needed material that was better levelled for them to be most productive.

Cut to this year when I have actual 6th and 8th graders and I decided to participate again.  My current motivation was less the problems themselves than the timing. We don't have any real contests to participate in during the Spring and I wanted to do one meaningful one for the kids who like doing them.  Since you have a testing window when you can administer the contest and it just needs a few computers, the overall experience is very low barrier (much easier than an AMC test).

Overall this year went much better. I split into two teams and we worked in the library. Both groups found answers to over half the problems and most kids stayed on target.  I had  few at the last 10 minutes who had reached their limit for the day which is not unexpected.

I'm once again trying to decide which problems to review as a group next week. I'm thinking we probably should only do 3 maximum so perhaps I'll bring a set and let the kids vote on which ones they want to see the most.

In the meantime and in no particular order here are my observations about a few problems I noticed and thought about while proctoring, now that its ok to discuss them:

Problem 17

Let a, b, c, and d be real numbers such that \( a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018. \). Evaluate 3b + 8c + 24d + 37a.

This one is phrased in such a way as to suggest non integer answers but just looking at it the temptation is to say what  if the two expression are completely identical and 3a = a^2, 8b = b^2 etc?  And if you compute the squares sure enough  \(3^2 + 8^2 + 24^2 + 37^2\) does equal 2018.

That's fairly easy for a problem at the end and its not a very satisfying method. So what I think they might have been aiming at was to subtract the two expressions and complete the square to get

\( (a-\frac{3}{2})^2 + (b-4)^2 + (c - 12)^2 + (d-\frac{37}{2})^2  = \frac{1009}{2} \)  The right hand side is 1/4 of 2018 so you can plug the original equations back in to get

\( 4( (a-\frac{3}{2})^2 + (b-4)^2 + (c - 12)^2 + (d-\frac{37}{2})^2 ) =   a^2 + b^2 + c^2 + d^2   \)

Then subtracting the right hand side again you get:

\(  4(a-\frac{3}{2})^2  - a^2 + 4(b-4)^2 - b^2 + 4(c - 12)^2  - c^2+ 4(d-\frac{37}{2})^2 - d^2   = 0 \)

Each one of those pieces is a difference of square for example the first one is \( (2a - 3 + a )(2a - 3 - a)  = (3a - 3)(a - 3) \) and if we set a to either 1 or 3 will zero out etc. You can do a similar operation with the other original expression and see that a = 3 for instance is the overlapping solution to both new equations. That gets to the original observation once you test the values in the original function. But it doesn't rule out alternate solutions where the pieces balance each other out in various ways.

Problem 16 

On  \( \triangle{ABC} \)  let D be a point on side AB, F be a point on side AC, and E be a point inside the triangle so that DE is parallel to AC and EF is parallel AB. Given that AF = 6, AC = 33, AD = 7, AB = 26, and the area of quadrilateral ADEF is 14, find the area of  \( \triangle{ABC} \)

A significant part of the difficult with problems like these is getting an accurate drawing from the description. That's something we can definitely practice as a group.

Assuming you arrived at the following:

You can deal with the easier part of the problem which is all about triangle area ratios. First lets remove the unneeded lines and split the parallelogram in half.

  •  First note ADF is half of the parallelogram and has an area of 7.
  • Then the area ratio of ADF : ACD is 6:33 
  • You then repeat this process: the area ratio of ACD : ABC is 7:26

Problem 13 

Suppose x and y are nonzero real numbers simultaneously satisfying the equations \( x + \frac{2018}{y} = 1000 \) and \( \frac{9}{x} + y = 1. \) Find the maximum possible value of x + 1000y.

My first instinct in these problems is to always remove the fractions to get:

$xy + 2018 = 1000y$ and $9 + xy = x$ Then just on inspection we have the 2 parts of the expression we want to simplify   \( x + 1000y = 2xy + 2027 \)

And we also have an way to isolate either x and y, I picked y, to plug them back in and get

$y = \frac{9}{1-x}$ =>  $x + 1000 \cdot \frac{9}{1-x} = 2x \cdot \frac {9}{1-x} + 2027$

That cleans up quickly to: \( 1000x^2 - 3009x + 2018 = 0\) and while the numbers are high the factorization is still not too hard (1000x - 1019)(x - 2).  So despite the phrasing which suggests an optimization problem there are only 2 solutions  for (x,y) and you just have to plug them in and compare to get the final result.

Problem 15

There are integers \(a_1, a_2, a_3, . . . , a_{240}\) such that \( x(x + 1)(x + 2)(x + 3)· · ·(x + 239) = \sum_{n=1}^{240} a_n x^n \). Find the number of integers k with 1 ≤ k ≤ 240 such that \(a_k\) is a multiple of 3.

At first glance this is a slog of a counting problem. Multiplying all 240 binomials together is impractical without a computer program.  So the first step I took was to look for patterns by doing the first few terms,

Ignoring x which doesn't change the coefficients:

$$(x+1)(x+2) = x^2 + 3x + 2$$
$$(x+1)(x+2)(x+3) = x^3 + 6x^2 + 11x + 6$$

I noticed a few things from this:

  • all the x+3n terms always added 1 more multiple of 3 coefficient than the last term. This makes sense the x term just shifts the previous result and then we add a multiple of 3 to it which doesn't change the value modulus 3 and finally we get one new multiple of 3 constant term at the end.
  • So grouping the terms looked interesting.  Note: there are 79 terms with  a multiple of 3 in them. So that's 79 coefficients with a 3 at the end as a minimum. 
  • Next: modular arithmetic seems useful here. We can simplify everything to mod 3 and not change the result so now we have x(x+1)(x+2)(x+3)(x+1)(x+2)(x+3) ....   I chose 3 rather than zero deliberately so we didn't lose any terms.
I also started simplifying the calculations letting x be an implicit place value system  and never carrying so

for example squaring the first term (x+1)^2  becomes (1 1)^2  = 1 2 1 

Somewhere around this point I also converted to balanced ternary and used  \( ( x+3)^{79} \cdot (x+1)^{80} \cdot (x-1)^{80} \)

That's convenient because we can mostly easily expand the two final terms using the binomial theorem mod 3. At this point you're left with multiplying two highly symmetrical 81 digit terms. 

As far as I  can see there's no escaping working out this product at least to a few terms before the pattern is visible but its a fairly simple process and you quickly see the result looks like (1 0 1 0 1 ... 0 1) with 161 terms and therefore 80 more zeros which are really coefficients that are multiples of three. That gives the baseline number of coefficients. So we add the two parts together to get 80 + 79 = 159. 


I went with another UWaterloo problem set this week: with a fairly approachable  number theory/lcm type problem.