Two Proofs and a Lie

I’ve caught the crud. It is either from:

  • My tiny toddler germ factory and/or her daycare
  • My teenage student germ factory

Your guess is as good as mine, but I’m home with a 100 degree fever, and so is my kid. I’m thanking my lucky stars that I selected a primary care doctor in the same building as my kiddos pediatrician. Nothing says family bonding like getting a nasal swab at the same time as your kid. Good news: we’re flu free. More good news: my kids napping so I get to blog.

Now, to the math. Last week I did an activity in Geometry and I promised @druinok that I’d blog about it. I called it Two Proofs and a Lie, its based off the Two Truths and a Lie structure that I first saw on Jon Orr’s Blog (he’s done a new posting on it recently which re-reminded me how much I love it!). What I love most about this structure is that you need not waste a time explaining what to do–its in the title, it says, “hey student, I’ve given you something that contains two true things and one lie…figure out which is which.”

We were on our second day of quadrilateral proofs. Students had figured out that most of the proof center around finding a triangle that helps prove the thing we want, prove those triangles are congruent, then CPCTC (i.e. their favorite mathematical acronym). I wanted to push them a little. But a doable amount. So I gave them the handout (here) and told them they had 5 minutes of no writing brainstorming with their peers. They were to discuss what it is they are being asked to prove, what was given, and how they think they might find that information.


Students were then told they could use whiteboard markers to collect their brainstormings into a proof rough draft. My kiddos don’t like the #VNPS for some reason (probably because chalkboards aren’t as ‘cool’ as whiteboards), but they LOVE writing on the desks with markers…so I have #HNPS (horizontal non-perminant surfaces).


This drawing is from one of my most reserved students. They never really talk in class, and always take a back seat to their peers. I was walking around and their group was 100% confident they had found the lie, and this student, my usually quiet one, goes “um. guys. I don’t think your right. I found the triangles like Mrs. White said. This ones true.”

The thing I like about making them brainstorm twice is they actually have to say what they are thinking out loud. I do believe that there is an added benefit to verbalizing what you’re thinking when it comes to math, but in particular with proofs. The biggest hurdle I’ve noticed with my students is forgetting a step in a proof because “they know its true.” Well yeah, its true, you know it. I know it. But you’ve gotta say how you know its true in a proof. We ran out of class time here. Students took photos of their white boarding work and we picked up the next class.

The next day students then wrote up, as a group, the two proofs that were true and peer-edited them before turning them in. I gave groups feedback, and we then moved, in the following days, to more individual proofs.


Groups working on their proof write ups.


Peer Editing the proofs


Proof Writing


Did I mention on day 2 of this my kiddo had an ear infection and couldn’t go to daycare so she spent the day in my class? 


My Digital Filing System

I had the pleasure of attending NCCTM this past week in Greensboro. It was a breath of fresh air to surround myself with teachers striving to improve our teaching practices–I just love having the opportunity to put myself in the learning seat for a while. Not that I don’t enjoy teaching, I love it, but I also love learning…which come to think of it is probably why I like teaching.

But anyhow, when it comes to teaching I do tend to assume that almost everyone else has a better handle on it than I do. I’m on year 6 and every year I have a list a mile long of things I wish I had done better/could have improved/need to improve for next year. The internet Gods gave me the Math Twitter Blog-o-Sphere (#MTBoS) when I was in grad school and their sharing of the happenings in and out of their classroom that have kept me afloat in my time teaching. My online math family keeps me bettering my practice and I love everyone of them for all they share.

At NCCTM we were recruiting newbies to Twitter and it was brought to my attention that some of my MTBoS family don’t have a good system for storing the wonderful things we have borrowed from our friends. I’d say I spend about an hour on Twitter in a day, mostly while I’m cooking dinner or waiting on photo copies to be made…and often when I’m procrastinating from other things. But my MTBoS family post really good stuff that I don’t always have time to read at that moment–I have a two year old after all, time is something I never have enough on–so I store it away for later. This way I can re-look through the goodies when its on my mind or I have time.

So here is my system:

Screen Shot 2017-11-04 at 3.57.12 PMI use Evernote to store all of my internet findings. It is free to sync across two devices, and you can pay for a premium account of you want more devices. Evernote is meant to be used as a digital notebook–I originally got it to try and keep myself paperless in grad school, but quickly decided it had a better use: Storing all the goodies I find on the internet.

So, I’m not sure if this is the best way to use Evernote, but its the way that works for me:

  • Download the Evernote extension in your web browser. I use Safari, so I got the extension here. This allows you to be one button click away from storing the site.
  • I also use the app for my computer, but that’s because I like its layout better than the web-based view.
  • So, let’s say you find an awesome resource you want to save. Click the WebClipper Extension (elephant icon) and the magic window of storage will open in the top right corner of your browser.

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  • You can then choose what notebook you want to store the website in. I’ll talk more about how I set up notebooks in a second.
  • Then you add tags, these are storage labels that you might use to find the webpage later. For instance, “Desmos”, “Linear Equations”, “Linear Systems” These are words that you can search through later on to re-find things.

So somewhere along the way I decided to organize my notebooks by standard clusters. So things like “Linear Equations”, “Quadratics”, “Trig Functions”,  or “Triangle Proofs”. There’s still evidence of my first filing system, it was by course, but things got far too cluttered for my mind.

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Now that things are organized by standard, if I’m gearing up for a linear equations unit, I just click on the linear notebook and I can see all of the goodies I’ve stolen borrowed from my MTBoS family.

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This is what my Linear Equations Notebook looks like. The middle column has small snips of everything that I’ve saved. The right column will preview any of the sites that I’ve saved. To go to the website I simply click the URL at the top of the application by the date.

Can you save other things to Evernote? YES! You can save screenshots, which is the easiest way to save tweets. When you save a screenshot you are given the option to add annotations like arrows, lines, shapes, you can blur out faces, and add text with the built in editor (which is also available for your phone in app form).

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(Sorry Julie, I couldn’t resist this screenshot hehe).

You can save PDFs, you can write your own notes to save. And my favorite feature is to add text to a page that I’ve saved. So let’s say I found a 3-act task I LOVE but might want to add a reminder to future-me that it requires computers, or extra materials for class. Its a mental reminder for “YOU NEED THIS STUFF BEFORE YOU USE THIS” feature.

I’m hopeful this helps you. I’ve been told I should write a proposal to chat more about this at NCCTM in 2018, but I thought I’d get a blog out in the meantime. Also…I really need to get better about long blogs. My 180 blog has been more successful this year as I LOVE to take photos 🙂


First Day Plans

Its the Saturday before school starts and I didn’t finalize what I was doing for one of my classes until well into my second cup of coffee of the morning. I’ve had geometry planned for a while. I was easily able to select The Marshmallow Challenge for them. Partially because I taught half of them last year and didn’t do a wonderful job of establishing group working norms, and partially because it is one of my favorite start of school activities.

Geometry Day 1: The Marshmallow Challenge

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For those of you unfamiliar with the task, you give your students 20 pieces of spaghetti, a yard of tape, a yard of string, and a large marshmallow. They have 18 minutes to construct, with their groups, the tallest free standing structure that they can. The catch is that the marshmallow needs to be on the TOP of the structure.

I like this activity for three reasons:

  1. It pushes students through the phases of group dynamics quickly. They move from the “oh hey, its kinda awkward working with these people I don’t know” to “hey, you, yeah, what’s your name? Ok [insert name] hold this pasta for a sec so we can tape it” to  “OH MY GOD WE ONLY HAVE 3 MINUTES LEFT WHO HAS THE MARSHMALLOW?!?!?!”
  2. I really get a feel for how my students think through a task. These towers have varied from elaborate to simple tri-pod-like.
  3. The debrief possibilities are endless. There is a wonderful Ted Talk by Tom Wujec that I use every year I do this challenge. Spoiler alert: Kindergartners are better than most adults at this challenge. Its a GREAT launching point for talking about the prototyping process and how we can mirror the practice in a mathematics classroom.

AFM Day 1: String Structures

This is the ideas I re-discovered after some of my morning caffeine boost. I had saved this image in Evernote filed under my “MTBoS” & “Group Work” tags”

Screen Shot 2017-08-12 at 9.00.32 AMI can’t remember what the exact instructions were for the construction, but I know what I am going to make them:

Your group has been given 2 yards of string. Your task is to create each of the shapes shown using only your group members’ hands to maintain the structure. Each group member must play an active roll in creating and maintaining the shape. 

Once your group has created a shape, call Mrs. White over to document your awesomeness and then start another shape.

So it goes without saying that I have no idea how this activity will go as it is my first year trying it. But I have hopes for how it will go. For most of my students in AFM they have, as one parent put it at Open House yesterday, “a fair amount of unpleasant mathematical baggage.” Something that feels overtly Math-y isn’t a very user-friendly start to the school year for them. So I want something that feels un-math-y but that falls into the Math category of geometry–I have found that my AFM students tend to have pleasant memories of geometry, so I’m using that.

I am going to give my students 20 minutes to try and construct as many of the shapes as they can. We’re then going to debrief and create some class group work norms. I then want to do a Talking Points activity from my morning Twitter Math Camp session, Talk Less, Smile More.

Things I left out of this plan because the 3rd cup of coffee is just now being consumed and my brain takes a while to wake up on the weekends:

  • My classes will be doing Sara VanDerWerf’s Name Tents because it is the most amazing “get to know your students and help the name-forgetting-teacher learn your names” thing I have ever found. I haven’t finalized the questions I’ll ask each day.
  • I will be sending my students home with one assignment: To complete a student info sheet (which includes preferred pronoun so I can add that to my roster) and log into the web-based tools we will use this year: Canvas, Desmos, DeltaMath, GoFormative.

Goals, The 2017-2018 Edition

So as I sit here at the kitchen table, listening to my husband read “The Going to Bed Book” to my little kiddo I have 9 more wake-ups until I greet my first students of the 2017-2018 school year. This year I’m teaching AFM (year 4 of this) and Geometry (first time teaching it) and I had a to-do list a mile long for the summer and I feel as though I only got through the first 200 yards. I’m freaking out a little. But when I start to freak out, I do what I think most teachers do, I make a plan.

So here’s my plan for my mental sanity as I venture in teaching a new course and re-vamping an older one:

  1. Go to bed early. To those of you who met me at Twitter Math Camp, I managed to Screen Shot 2017-08-05 at 6.26.47 PMstay up until 10:30 every night so that I could be social and meet the lovely internet people who have helped me along this teaching journey. But that’s WAY past my bed time. I’m the human embodiment of  “Grumpy Cat” when I don’t get my sleep. Don’t believe me? Well my daughter can make the “Grumpy Cat” face, she had to have learned it somewhere. I’m guessing it was from me. My goal is a 9:30 bedtime. Last year I made the misguided goal of having 30 minutes of device-free time before bed, that didn’t happen. And I’m okay with that. I like my Twitter time.
  2. Instructional Routines. Find the ones I like and stick with them. Okay, so I’m REALLY bad at being a little ADD with my teaching practices (and blog posts, bear with me). Last year I feel into some, in my personal philosophy of teaching, bad practices of teacher-centered routines. It was a lot of “I Do, You Do, We do.” I fell into that pattern because I was sleep deprived (see goal 1) and I had 3 preps. So I felt like 2 of them never got my full attention at any given time of the year. I could make excuses. But I’ll just say, I know I need to do better to be in my happy teaching place. Designing a new course from scratch I think is going to help keep me accountable for establishing these routines. So I have high hopes. Here are my chosen routines:
    • #VNPS, Vertical Non-Permanent Surfaces. Well, in my classroom, with all its windowed glory, the vertical part is not going to work. So I’ll be using #HNPS. Horizontal Non-Permanent Surface. I bought shower board this week and had them cut into 2×3 foot sections and will get to taping the edges next week when the “OMG ITS ALMOST THE FIRST DAY OF SCHOOL” stress hits me and I need a mindless task. The goal here is to make students mathematical thinking visible and a group effort.
      • A KEY subsection here is visibly random grouping. Students need to know that there is no rhyme or reason to their group assignments. I am going to do new groups on Mondays because I know I’ll never learn names if I can’t get students in the same seat for at least the first week of school.
    • Debate Structure. I went to Matt and Chris’s TMC morning session, “Talk Less, Smile More” which was on using debate structures as a method of increasing student talk and engagement in a task. I mean, have you met a teenager who doesn’t like to argue? Yeah. I didn’t think so. The cliffs notes version make a student pick a side (or give them a side) then have them provide their justification of the form “My Claim is______, and my warrant is _____.” I was pleasantly surprised how having that sentence structure made me feel less introvert-y and willing to participate in a room filled with strangers. I was sold.
    • Desmos Activity Builder. I’m holding myself accountable to making/using more of these. The students always love them. We have great conversations. I think I just get caught up in trying to make them look “Desmos Bank” worthy. This year I’ve vowed to get over that. Just make stuff that works for my class. I’ll make them fancy and pretty when I have time.
  3. Stay on top of my Web-Based Things. Yeah, okay that sounded vague. But I don’t know how else to describe making sure I make Delta Math and GoFormative homework well in advance so I don’t get stressed, and keeping my notes and note keys up to date on Canvas (our online learning management service). I was bad at this last year. So I have structured an hour into my lesson planning block–I have from 11:15-1:30, I know I’m lucky to have that time and I need to use it better.

And that’s it. I read somewhere you shouldn’t try to change more than 10% of your actions at any given time to ensure success. I’d venture to guess this is more like a 15% change. I’m keeping my guided notes problems from years past. I just plan to make them debatable instead of “here you do this.” If the problem doesn’t feel debatable, I’ll make it #HNPS. If I’m feeling super Desmos-y I’ll make them an Activity Builder. I’m not re-inventing the wheel, just giving it some new tread. Or maybe some new rims? If I’m feeling like an overachiever I’ll go for the spinner rims  upgrade and use that Desmos Computational Layer magic and “live dangerously.”

Making the Leap: From Lurker to Participant

I just recently returned from Twitter Math Camp, a grassroots professional development run by members of the Math Twitter Blog-o-Sphere (#MTBoS). It was the most UNBELIEVABLE week of my life. I learned more than I could have hoped, and finally got to #InRealLife meet my internet friends and family. While I was at #TMC17, Dan Meyer posted about Retiring the #MTBoS Hashtag, and I had a lot of emotions. Most of them resembled feeling hurt. This is my internet family. I have always felt a part of the warm and loving mathematical community that he helped shape and mold. His posting was edited and amended over the course of the week, but I still felt…something resembling pain. I think I know what Dan was trying to start, a conversation about making the #MTBoS more user-friendly, but it stung a bit. I’m glad the conversation’s been put out there. And I see our community doing the work to increase our user-friendly-ness.

Before I continue, let me say:

I am a proud member of the #MTBoS. I found Dan Meyer’s blog in grad school, and the 3-act task structure he introduced me to changed my teaching for the better. While I found Dan in 2012 and #MTBoS in 2013, I was a (proud) lurker. I’d tell every math teacher I met about this WONDERFUL group of math teachers who put their classroom content and mathematical souls out on the inter-web for all to see.

In 2015 I boldly used the #MTBoS hashtag for the first time. I started to transition from lurker to participant. But even then, in the prime of my transitional-lurking days, I mostly re-tweeted, and liked, and asked questions offering up little to nothing in return. But that’s what I love about this community. I asked, and asked, and asked, and my #MTBoS family stepped up and gave all they had without critique or judgement.

I’m proud of my lurker days. Lurking was a necessary part of my development as a teacher. I wasn’t ready to put myself out on the internet in 2013, or 2014, or damn, even in 2015 really–I wasn’t ready for that level of vulnerability online.

To be clear: That feeling of not being ready to proactively join the #MTBoS conversation is not to be confused with feeling excluded or unwanted from the community.  Although I hear some people did/do feel that way, and to you all who felt/feel excluded I am deeply sorry for any part I may have played towards your feeling excluded from our mathematical family. This family thrives on new additions. New voices. New opinions. We need you in our family, our community, our conversations, and our classrooms.

Twitter is a scary medium to put yourself out there, and its even scarier when you don’t really know #InRealLife the people you’re talking to. But those same aspects that make Twitter scary–talking to those strangers on the internet–is what can make it the best professional development opportunity. Those strangers are online 24 hours a day, 7 days a week, 365 days a year to give you feedback if/when you ask for it.

Some things I learned as I moved from lurker to participant that might help people who are still proudly lurking. No rush. Move at your own pace. But here’s some things I discovered along the way that might help with the transition:

  • When in doubt. Comment on cute puppies, cats, tiny humans, or what ever else floats your boat. I remember this thread like it was yesterday, it was the first time Meg and I talked about dogs! Now, we chat about our fur babies at least once a week. Yes, there is a hashtag #DogsOfMTBoS!:Screen Shot 2017-08-01 at 7.56.50 PM  Screen Shot 2017-08-01 at 7.56.59 PM
  • Talk about something math-related that may have NOTHING to do with teaching. @cheesemonkeySF, @anniekperkins, and @veganmathbeagle  are always posting about really, REALLY awesome math art. I’m currently memorized by the Celtic knot designs they are doing right now. If I wasn’t so into my 10:30pm at the latest bedtime, I would have taken more time at #TMC17 to learn how to draw them. Maybe next year!
  • Just adding #MTBoS or #iTeachMath(s) on the end of a tweet isn’t (always) enough to get feedback. There is no #MTBoS high council who patrol the internet moderating comments. If you are looking for specific feedback, tag specific people. When I needed algebra 2 help, I looked to the #Alg2Chat and found Julie ( @jreulbach, and Meg (@mathymeg07) I tagged them when I had a specific question, and I heard back. Find those who teach your content and direct questions their way. You will hear back, and if you don’t, tweet me. I’ve taught everything from Algebra 1 to AP Calc. I’ll do my best to help.
    • Chats to look into:
      • #MTBos and #iTeachMaths, obviously.
      • #GeomChat
      • #Alg1Chat
      • #Alg2Chat
      • #PreCalChat
      • #msmathchat
  • Set goals and make them public. I told Meg I’d be at TMC17. And it happened!

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So with that, come on and tweet with us. Use what ever hashtag you like. I’m going to stick with #MTBoS because to me it means family of nerdy math people who happen to use the mediums of Twitter and Blogging to hold us over until we can meet up, once a year, for nerdy game nights and talking mathematics until WAY past our bedtimes.

M&M Lab

Advanced Functions and Modeling is an alternative to PreCal course. And this year I choose to move the Descriptive Stats unit from the very first unit in August to the first unit of the second semester in January. Our Stats unit is very simplistic as it is meant to be an introduction of sorts. Tt contains the same content from CCM1: Measures of center, calculating 5-number summaries and outliers, creating histograms and box plots. We choose to have the unit focus on arguing with data rather than purely analyzing data.

Arguing with data, for the purposes of the class is when students are given data, a side-by-side box plot, or multiple histograms on the same scale and then they need to:

  • Make a claim–students need take a stance for the data given. We modeled these statements to reflect the thesis statement construction from their US History class (which most students are in).
  • Provide three supporting statements for the claim. Each statement should be data driven and assist your claim.

The Lab was inspired stolen from an image I found on Twitter. I can’t find the source, most likely someone in #MTBoS.


And I figured this was a lab that could be easily recreated for my students. I, with the help of my loving husband, created “Mystery Morsel” packs. We created packs made from large index cards that we put 5 M&Ms in and sealed. We made 7 packs for each type of M&M.

Students were tasked with identifying the “Mystery Morsels” by using their weights. Each group was given a control baggie of M&Ms to come up with a baseline for each type. They shared their data on the board for weights of their M&Ms and then proceeded to weigh the “Mystery Morsels”:

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Was it a perfect lab? Heck no. I think it was far too easy and there are many things I want to change for next year. Several kids noted that “it was obvious” which M&M goes to which pack because the weights are varied. One student mentioned that it would be “evil if Mrs. White hid skittles in here somewhere. I bet they weigh about the same as the plain ones.”

Which got me to thinking…skittles…yes. And Reese’s Pieces. And maybe one random packet that has 6 M&Ms instead of 5.

My rational for this “trickery” is to have some of the box plots overlap in the range of the data. So students actually have to use their knowledge of measures of center/variability to rationalize their decisions for which of the “Mystery Morsels” belong to which M&Ms and which are “other” candies.


Lab handout

Week 1 in Review

This year marks my third at UNCSA and I have never looked forward to, and dreaded, the start of a school year so much. I’ve had the pleasure of staying at home for the past six months since the birth of my daughter. That being said, I am not cut out to be a stay at home mom. I’m not organized enough, I’m not good at keeping my brain engaged and working, and I developed a bad case of cabin fever. So in one way I was REALLY looking forward to the start of the 2016-2017 school year. But I dreaded the start of the year because I have felt behind since August. I didn’t get the lesson planning done during maternity leave I (naively) had promised myself I’d do before my daughter started daycare. I haven’t spent the time I feel that I should at home lesson planning because I’d rather be holding my girl and planing peek-a-boo (do you blame me?). And as luck would have it, she got an upper respiratory infection the week school started, so I’ve spent the past few nights up at all hours rocking her to sleep, which left me no choice but to be highly caffeinated to survive the teaching day. I have never felt so shaky at the beginning of the school year, but I think I did do a few things right (or at least I am calling them a win) from my maternity leave to-do for the 2016-2017 school year:

Get better about learning names:

I used Sara VanDerWerf’s Name Tents to aid in both the learning of my students names, and in getting to know them a little better within the first week of school. It was the best decision I could have made, I may never have to try another way to learn names. It was super awesome to get the chance to have a private conversation with each student. They kids responded to the prompts:

  • What do you want me to know about you?
  • Give me your 6 word math memoir.
  • What’s your favorite thing to do outside of your art area? (I teach at an arts school)
  • Doodle me something about yourself and explain the significance.
  • What is your most vivid mathematical memory?

My favorite was the doodle, you’d think that in a classroom with a few visual artists and a ton of dancers, musicians, and actors that the kids might be shy on doodling. Nope. I drew them a picture of my stick figure family on the screen with the prompt, so I set the doodle-bar kinda low intentionally to set their minds at ease.

Don’t go Over the Syllabus in Class

Some where along the line, someone told me to go over the syllabus on the first day. That it sets expectations/demonstrates norms/ya-da-ya-da. I hate syllabus day. Even last year’s attempt at making it bearable (each slide had an internet meme and/or GIF), you know the ones:

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But it was painful, the kids stopped listening 3 slides in (once I was done talking about my story and started talking about “class rules” kinda things). So I ditched it. My syllabus lives on Canvas, our online learning platform. My students are all sufficiently good enough readers to handle reading it on their own–with the exception of a few ELL students who I made time in the first week to talk to about the syllabus to ensure they were up to speed. So my first night’s HW assignment was to read the syllabus and to take my online Student Information Quiz (in Google Forms format). Students then logged into Canvass and posted three questions they had regarding the syllabus in a discussion forum, and answered one peer’s question. I addressed any further concerns on the second day of class and it took less than 10 minutes. Win!


Focus on Group Dynamics, Not The Math

I know, I know. “You’re supposed to do Math on the first day” to set the tone for the year. Yes, I agree, but I wanted the focus of the first week of my class to be on the group work dynamic so that when we step up the math, they are already comfortable working in groups. The first math problem I did with my students I sawn on Fawn Nguyen’s blog that she got from Don Steward:

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The problem was more successful as a day 1 task in my PreCalculus and Advanced Functions and Modeling classes than in my Algebra 1 class. But they all had some pretty stellar math talk going on and eventually (with differing levels of hints) got the final answer. On a related note: America needs to get on board with the metric system.

Day 2 we did a task I first read about in Designing Group Work and Sarah Carter was so kind to provide already made up on her blog: Rainbow Logic. The idea is that the students use strategic questioning to infer the color pattern on a peer’s tic tac toe board. The kids got WAY into it. I should have taken pictures, but several groups were convinced that their facial expressions were giving away information, so they hid behind the file folders I gave them to separate the “game keeper” from the players. We did four rounds of the activity. Three of the rounds were with 3 colors and the last round was with 4 colors. So far my students record for number of questions to get the correct color combination was three.

  • They asked for the first row’s colors
  • Are there more than two of each color in any given row?
  • Is the center color blue?

I didn’t witness the whole thing go down. But I am to understand that there was an educated guess on one square placement that yielded success.


It was a good first week. And it only took me until half way through week two to get around to reflect on it…