Wow, excellent post!
For the floating ball problem, I had the same intuition as Bucky: A floating object will displace its own weight in water, so we can replace the ball with an amount of water that exactly fills in the hole created by the submerged part of the ball. So the box is balanced no matter where you move the ball to.
"Thinking Physics" by Epstein is an excellent source for puzzles like this, where you identify the relevant physical principles to give qualitative answers as opposed to doing tedious calculations.
Thank you for the write up! Matthew Walker was also on The JRE Podcast and has a Talk at Google. Well worth checking out.
Some of my experiences tutoring math over Skype:
One-to-one tutoring is superior to one-to-many tutoring.
One-to-one tutoring can be structured as a conversation with lots of back and forth and keeps student participation high. It also means the teacher can adjust their teaching style to that particular student's level and personality.
One-to-many tutoring is more of a performance on part of the teacher and is quite different. The students are much more passive in this situation. I don't know how to do this well.
The most important thing is to get t... ,,,,,,,,,
I never looked at the notes I took [...] I did notice, however, that I remembered things better when I wrote them down, so for a time my plan was to simply take the notes and forget about them in my binder.
Beethoven did the same:
Beethoven left behind an enormous number of sketchbooks. Yet he himself said he never looked at a sketchbook when he actually wrote his compositions. When asked, "Why then, do you keep a sketchbook?" he is reported to answered, "If I don't write it down immediately I forget it right away. If I put it into a s...
I'll add another supporting quote. Mathematician Niels Henrik Abel says of Gauss : "He is like the fox, who effaces his tracks in the sand with his tail."
A counter-example to this sleight-of-hand behavior is Euler. As related by Polya here: https://archive.org/stream/InductionAnd AnalogyIn Mathematics1 #page/n109/mode/2up
(markdown bugs up the underscores in my link)
I wonder the same! It's almost like a reverse version of Four-Leaf Clovers. (I'm not a biologist, just a dude with a camera.)
40 minutes later, I was able to catch three of them in one picture. (One is right by the flag reflected in the water, the second jellyfish is below it, and the third is below that again, partially hidden by the dock): https://imgur.com/a/rrCff
One of the rings in the center jellyfish looks slightly elongated, almost as if two rings have melded together, but I have no idea if that's even possible. For all I know it'
Apropos jellyfish symmetry, I saw a couple of trilaterally symmetric jellyfish in the Flensburg harbor this summer. Here's a picture of one (center right): https://imgur.com/a/m5tdh
Compare Helen Keller with Ildefonso who was deaf and mute, and didn't learn language until age 27:
" But the interesting thing that he said is that he can’t even think that way anymore. (Music.) He said he can't think the way he used to think and when I pushed him to ask about what it was like to be languageless, the closest he ever came to any kind of an answer was exactly that. I don't know, I don't remember. I think differently now. " - Radiolab episode on words