Welcome to the Playful Math Carnival, 155th edition!

155, tell us your secrets.

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 calendar,

learning 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**

Andrew Fenner made a

hundred chart game in KnowledgeHook. (Free account but you have to log in to see it.)

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**

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.

**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.

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.

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.

**Tik Tok?**

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

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

**Off Ramp**

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!