### Small Geometry Puzzle

Feb. 13th, 2015 11:11 pmI really like this puzzle. And I'm sure that there's a simpler proof than the one I'm using right now.

Puzzle:

**( Solution )**

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I really like this puzzle. And I'm sure that there's a simpler proof than the one I'm using right now.

Puzzle:

On top of that, I've been abusing the program in a way - using it to draw entire scenes in addition to communicating the word. :) There's a limit to how much "ink" you can use (which prevents me from spending forever on a single picture). And there's also an immediate objective and audience every time I draw. It's been a pretty perfect solution.

Here are some examples, and more of my pictures are hidden below. And if you think this sounds interesting and want to try it, maybe invite me to play a game with you (my username is "ch12cooh"), or I can invite you if you let me know you're intersted.

app link

Sushi, Mars, Sphinx, Hayride, Ocean...

And here are a few pictures that other people on Draw Something drew for me:

The ribbons the students (and instructors) are playing in are sequcens that start with any number of the students' choosing and then progress using the following rules: If the latest number is even, divide it by 2 to find the next number. If the latest number is odd, multiply it by three and then add one to find the next number. The magic is that, so far, every initial choice of a number eventually leads to the same behavior - eventually, the sequence trickles down to the cycle 4-2-1. But it's unproven if this is true for ALL numbers... it just works for all of the billions tested so far. :D And sometimes it takes a while for the sequence to get down to that cycle. For example, our TA chose to start with the number 231 which has a sequence that goes for 127 steps before cycling.

Collatz Calculator: http://skanderkort.com/collatz_conjectu

I want to try a similar session out with an older group and combine it with logic tables of pairity. :) And I think this might be the last part of an "infinity" curriculum that I've been trying to piece together! :D

And there aren't any spoilers in the details below, just big-picture writing/style observations. :)

:) I finished the Mistborn series a while ago, but then had a rough and busy couple weeks. However, wrt the end of Mistborn: IT. WAS. PERFECT! Really in pretty much all ways I can measure, it was such a satisfying conclusion. The plot and world-building are impossibly well crafted and last quarter of the 3rd book is one ridiculous climax atop the next, paying off

MoMath has been very very full of school groups for the past few weeks. I've been teaching 5 or 6 classes nearly back-to-back every day. Mainly Cryptography though, which is my favorite of our three classes.

But I'm finally settling into the 'routine' of this, and re-finding the time and energy to do other awesome projects at home on the weekends. :)

Way back in January, a bunch of friends sent me amazing pictures... and I got hosed and hoarded them all to myself. No more of this. Here are the results (my favorite pictures) from week 2 (of 4) More soon.

Category 1: ID -- Can you ID the subject matter of each of these shots? (hints behind the cut)

Category 2: Sometimes the subject matter is simply awesome

Category 3: Beautiful Locations

In real life, who has the power? - the person splitting the money, or the most stubborn and greedy player? Why? How can this question be described rigorously mathematically?

The Paradox

I'm really thrown by these puzzles/'paradoxes' and I feel like there's got to be a good theory somewhere to explain what's going on. I also think these questions are practically relevant since they suggest that things like mutinies might have a logical basis, whereas simple economic theory frequently predicts that the masses will just accept their short-changed fate.

THE PROBLEM

I'm currently scheduling interviews for teaching positions in a summer camp (Junction, an MIT ESP program) I talked with the interviewers and made a schedule of all of the interview slots during which we can convene for interviews. Now I need to email a bunch of prospective teachers and somehow learn enough of their availability to schedule them each for an interview. I'd like to do this without too much email correspondence back and forth since it's a lot of people to keep track of. My current plan is a bit complicated and something less complicated or technology that would help with this would be very appreciated.

THE PROBLEM MORE RIGOROUSLY

There are 35 good interview slots and another 16 ok slots. But for convenience, I'll simply say there are 51 slots. I'll call this set S (interview slots). S is a finite set of non-overlapping hour long intervals between February 21st and March 5th. They are either from an hour to an hour (ex:5pm-6pm) or from a half hour to a half hour (ex:5:30pm-6:30pm). They are distributed irregularly within the range of Feb 21st and March 5th. 20 teachers currently need to be scheduled for interviews, I will call the set of teachers, T. And each Teacher, t_i, has a subset of the interview times that they would be able to make, A_i < S. What is an efficient way to get information from each teacher about their A_i in order to find a unique interview time for each teacher? To further specify 'efficient', let's say that I don't want to ask for more than 10 bits of information in an email, and I want each teacher to need to send me as few emails as possible.

An algorithm that is communication efficient but not always the best is ok (EX: normally requires 2 passed emails with a handful of exception cases where 5 emails may be necessary). However, in this case, know that the trend in my past experience are that some people are very busy and some people are very flexible.

CURRENT PLAN

Right now, I'm planning on emailing them all a google form that asks for a bunch of basic info about their class. And, at the end of the form I would ask them to select a subset from a list like this:

Morning of Sunday Feb 21st

Late Evening of Sunday Feb 21st

Early Evening of Thursday Feb 25th

Late Evening of Friday Feb 26th

Where they select an entree if they are available for most of that time range.

I define each time-range of day as follows: Morning (9am-12:30pm), Afternoon(12pm-3:30pm), EarlyEvening(3pm-6:30pm), Late Evening(6:30pm-10pm)

I would then email each person two or three time slots to choose between within one or two of their specified time zones. And they email me back a final time selection.

So:

1) They get an email from me

2) They fill out a form

3) I send them a couple times to choose from

4) They pick a final time

But, honestly, this feels way too complicated! So a better method (from either mathematical or practical experience) would be awesome!

THANKS!