## 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)

#### Curriculum

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.

#### Contests

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

• MathCounts
• Purple Comet
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

http://mymathclub.blogspot.com/p/2017-2018-topic-map.html

## Friday, June 8, 2018

### More connections

I've been thinking alot about polynomial deltas recently. See: http://mymathclub.blogspot.com/2018/05/polynomial-deltas.html.  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    1            3
4           2
2    5            5
9           2
3   14           7
16
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.

Notes:
• 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:  http://mymathclub.blogspot.com/2018/05/522-chaos-mod-arithmetic.html   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.

• 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.

[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:  http://legacy.mathcircles.org/GettingStartedForNewOrganizers_WhatIsAMathCircle_CircleInABox

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.

P.O.T.W.

This geometry puzzle is actually a bit harder than I realized:
http://www.furthermaths.org.uk/docs/FMSP%20Problem%20Poster%205.pdf

## 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: https://en.wikipedia.org/wiki/Carmichael_function 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