## Saturday, April 30, 2022

### Playful Math 155

Welcome to the Playful Math Carnival, 155th edition!

Via Pat Bellew, 155 is the sum of the prime numbers between its smallest and largest prime factors, 5 and 31. 5+7+11+13+17+19+23+29+31=155. How would you go about finding more of these? What would you call them? Pat also notes that 155 is the number of primitive permutation groups of order 81. Which is odd, because it is more than double the number of groups for any order less than 81. And there's not another larger (than 75 even!) until you get to order 256 (which has 244). Do 81 and 256 have anything in common?

Wait, 5 and 31? That means 155 is semiprime. What is the previous and what is the next semiprime? (They're both even...) Are there more primes or semiprimes smaller than 100?

The coolest thing I found is that 155 is a toothpick number. You start with a toothpick, then add a perpendicular toothpick anywhere there is an exposed endpoint. Here is 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67. How many more steps to 155? Is it a fractal? Is it a cellular automaton? Mathematicians have also studied T(n)/n^2. Does it have a limit? Does it have an extremum? Here's some GeoGebra to make your own.

155 is also a generalized pentagonal number. The pentagonal numbers have a rule n(3n-1)/2, usually for n =1, 2, 3... , giving 1, 5, 12, 22, 35, ... But there are also positive outputs for negative integers, 2, 7, 15, 26, 40 ... which pleasantly fit between the usual pentagonal numbers. What patterns do you notice? Which negative number gives 155? I've been trying to think about how to visualize these negative pentagonals, to no avail so far. Have you got any ideas?

Maybe the toothpick was a little too crazy of a visual patten? Here's one I was trying to make to have 155. Did it work? If so, which step? Fawn always asks for the 43rd step... what's that? Is there a rule? What if step 1 had -1 square, what would the rule be?

I also found this pattern over at OEIS from Ilya Gutskovskiy. Which step is 155? How would you write the rule? What is a Fibonacci polynomial? From where did that question come?

On to the goodies...

Blogger of the Month
Jenna Laib is killing it. Creator of Slow Reveal Graphs, she has so much good writing on so many different topics, it is amazing. For example, THIS MONTH, planning (with a great pattern/multiplication activity), the Ramadan calendarlearning progressions,  mathematizing children's literature plus part 1 and part 2 examples. In addition, she edits the Illustrative Math blog, where she also sometimes writes gems like this on instructional routines Plus Slow Reveal Graphs, which just this week included How Loud is Too Loud?, Amazon Worker Injuries, and Australian Housing.

Elementary and Middle
Math for Love shared their 40 Face Puzzle. 100% will try, as I've loved the 100 Face activity, too.

Brian Bushart got playing Heads and Tails, a game/probability exploration.

Karen Campe wrote about special number pairs in math. The game I love adapting for these is Go Fish. For example, my preservice teachers were playing 1s Go Fish with some fraction cards they made with 4th and 5th graders. (2 cards each of: ½, ¼, ¾.⅓, ⅔, ⅙,⅚, 1/12, 5/12, 7/12, 11/12, one choice or can make two more different 1/2s, or a 0 and a 1.) I also made these fraction card blanks, but they might be more middle school...

Not this month, but there is a collection of tiny elementary math games here on this blog. Pointed for specific content, but low effort, low materials. As wih the fractions above, I love playing them with student made cards.

High School and Beyond
Henri Picciotto shared a blogpost from Liz Caffrey using his Lab Gear for algebra.

Deana Sample shared a fun bodyscale similar triangles activity.

Matt Enlow shared his progress on a crazy problem cutting up spheres to get different surface areas.

Also 3D, Sophia Wood shared her learners' work making nets for some interesting polyhedra in Polypad. (Which lets you fold them! Select all the tiles in the net, and a fold option appears. Select a polyhedron and an unfold option is there.)

Erin and Taylor, two of my seniors, put together a sweet 1 week graph theory unit for high school, which ends with a math game built on some pretty cool discrete ideas.

Mathigon shared their timeline scavenger hunt, using their excellent timeline of math and mathematicians.

Dave Richeson investigates Möbius strips with zippers with his learners.

James Propp applies proof by contradiction constructively in this month's post.

Math Art & Puzzles
Melynee Naegele sent the hexaflexagons from Sarah at Math Equals Love. These are always amazing! Sarah is also the queen of classroom puzzles, so check them out while you're over there.

Margie Pearse collected a bunch of math puzzles for May. (Gdoc)

Via James Propp and Daniel Kline, the Jumping Julia puzzle

Speaking of puzzles, Ms. Messineo sent Justin Aion's pride in solving Will M Dunn's puzzle. Feels like some kind of planar Ramsey Theory problem... Keep reading, the #mtbos discussion was pretty cool.

Patrick Vennebush wrote & joked about I Don't Know Puzzles.

Obviously I love using Polypad at Mathigon. Well they're having an art contest! For the under 18 crowd, but I'm planning to go gawk. HT Sophia.

Speaking of art, Paula Beardell Krieg sent Celeste Bancos' Origami Pockets post, which also had some great informal measurement investigation and what if thinking. Paula has been blowing me away with her #mathsartmonday tweets, like this one.

Lee Trent was playing with fractal cats. Fracatals? Not her first...

Speaking of tumblr, this poster described this viral video as stochastic continuous nim. Spot on.

Tik Tok?
Howie Hua is the king of math TikTok. Check out gems like his mixture puzzle.

The undisputed master of math tech, Steve Phelps is there.

Ms. Callahan is the funny math teacher.

Math Letters is shooting for a Math with Bad Drawings vibe for TikTok.

But there must be more! Help us find them...

Off Ramp
Karen Campe reminded me to promote Ben Orlin's new math game book, the epitome of playful math. I am loving it. Somehow it's even better than I expected. Karen also pointed out a pretty sweet hexagon tessellation at La Guardia of all places, so you know she has an eye for fun.

The previous Playful Math Carnival was at Denise Gaskins' blog, the founder of the carnival. Be sure to check her site weekly for the Math Game Mondays which are only up one week! Other goodies, too, though. Next up is at Nature Study Australia.  Contact Denise if you're interested or willing to host. It really impresses me every time I do just how much good stuff is out there.

PS. I've been working all year with Xavier Golden (yes relation) a preservice art teacher on a math graphic novel. And we're starting to see some inked and colored pages... I'm so excited!