Caleb Shor

enjoyed adding up squares in squares?
might adding up enjoy cubes in cubes?
might enjoy ... more???

here is a lil PROMYS for Teachers collection of fun!
courtesy of & curated by @cshor.org:
drive.google.com/file/d/1ysxU...

cc. #iTeachMath ♾️
bcc. #MathSky 🧮

Me: What is the smallest group whose order is a power of 2 that has no normal subgroup of order 2? It: Every nontrivial 2-group has a nontrivial center, and any element of order 2 in the center generates a normal subgroup of order 2. In particular, by the class-equation argument, every 2-group has a normal subgroup of order 2. Hence there is no (nontrivial) 2-group whose order is a power of 2 and yet has no normal subgroup of order 2.

FWIW, I just typed this prompt into math-gpt.org and it answered the question correctly.

(There appear to be a few math gpt websites. That’s the first one that appeared when I searched.)

Looks to me like you negated both the numerator and denominator (instead of just one of them) in the very last step. Otherwise I buy it.

Wow, this is a disaster.

Thank you for sharing tonight! And on such short notice. Always great to see you. I like the way your brain works.

I like this approach! As I read it I was going to mention that it’s in the pcmi materials. And then I finished reading what you wrote. Ha!

Another PROMYS for Teachers workshop tonight. There were some really fun show-and-tell presentations:

* Shapes of various shadows of polyhedra.
* How to see every positive integer coprime to 10 has a multiple that consists of all 9s (or 1s?).
* The existence of “printer errors” like 2^5 9^2 =2,592.

How to Reverse Declining History Major Enrollment Numbers, Which Are All the Faculty’s Fault How to Reverse Declining History Major Enrollment Numbers, Which Are All the Faculty's Fault

"We don't like to point fingers, but the History Department's drop in enrollment is totally the faculty's fault."

Hadn’t seen that! Thank you for sharing it. You have some really great MO posts.

The problem is quite doable without any knowledge of Legendre symbols btw.

and sending you stuff is on my to do list.

Ha. That was a fun read. There’s actually a lot of good stuff in there. A lot of handwaving too — not sure it’s providing a ton of insight. The key is why that sum of (n/p) * ((n+1)/p) is -1, which it dodges explaining. I’m not sure that’s a “well-known result” as is claimed.

I like it!

Err, squares mod p. Blah.

I like the quiz very much.

And I will share materials. Seriously. Previously I had the excuse that I was traveling. Now I’m not. But my computer is waaaay over there.

Here’s a problem I saw recently. For p prime, how many integers n in {0,1,…,p-1} have the property that n and n+1 are both squares?

I really like that fact about n dividing a number that’s all 9s! It’s a good one. I often put that on the final exam when I teach number theory.

Pretty sure! Otherwise the whole state would be shut down and we’d be open. (Kind of like on Presidents’ Day.)

Or sin x / n = six.

I think my university might be the only one in the country that was open on Presidents’ Day and closed the day after Presidents’ Day.

That seems reasonable. Are your students using it as well?

I haven’t seen any compelling reason to like it for the math I teach or do.

The main thing I’ve seen is students using it to pass off its “work” as their own. Maybe it’s good for certain subjects, but eg AI-produced proofs in an abstract algebra course are hot garbage.

I see a second player win. Player 2 (P2) selects the same 4 rows as player 1 (P1). Call those rows A, B, C, D.

If P1 makes a move in one of rows A and B, P2 makes the same move in the other row. Same idea for rows C and D.

This way, whenever P1 has a valid move, P2 does too. Fun!

This was fun. The extra helpful part at the end made me laugh!

Teacher Application Form – PROMYS | Program in Mathematics for Young Scientists PROMYS Boston University Dept. of Mathematics 665 Commonwealth Ave., 6th floor Boston, MA 02215 USA

The application for this summer’s PROMYS for Teachers program is now available. promys.org/programs/for...

General info about PROMYS for Teachers is at promys.org/programs/for...

Have any questions? Please get in touch!

#mathsky

And actually, I should mention that I’ve become very interested in units in cyclotomic rings (Z[w_n]), where w_n is an nth root of 1.
(Maybe we talked about this briefly last summer, I think the day that we were in a room different from our normal work room?)

It is of interest to me! Thank you for sharing.

Those are great questions!! ✅✅✅

Yeah something like that! (Bit more interesting if the number of units isn’t just a power of 2.)

Ooh, fun!

If n is prime and you can compute the cycle length of 1/n, I suppose you could use that information to determine whether or not 10 is a perfect square mod n. (?)

Seems like a fun job.