## Tuesday, May 24, 2016

### My third post on Gathering for Gardners

(Workshop for women in computational topology in August: see here. For a post about these kinds of workshops see here.)

(I have already posted twice on stuff I saw or heard at the Gathering Conference here and here,)

Meta Point- At the Gathering for Gardner conference I learned lots of math (prob more accuarte to say I learned ABOUT lots of math) that I want to tell you about which is why this is my third post on it, and I post more.

The pamplets of  Lewis Carol: Games, Puzzles, and related pieces: This was mostly puzzles that are by now familiar, but one new (to me)  struck me: an aloof word  is a word where if you change any one letter to anything else then its no longer a word. I think aloof is such a word.

Some talk don't know which one was about the Piet Hein Egg, also called a superegg. The talk (which differs slightly from he page pointed to) said it was a solid whose surface has the equation

(x/a)2.5 + (y/a)2.5 + (z/b)2.5

and its an egg which can stand on its end. (Note the x/a,y/a,z/b- that is correct, not a typo).
(Personal Note: Piet Hein invented Soma Cubes which is a puzzle where you put together 3-d pieces
made of small cubes into a large cube or other shapes. I learned about these in a Martin Gardner column and bought a set. I was very good at this- I put together every figure in the booklet within a week. This was the ONLY sign that I was GOOD at math when I was a kid, though there are many signs that was INTERESTED in math. About 30 years ago my girlfriend at that time and I went to a restaurant and there was a SOMA set on the table, assembled into a cube. I took it apart and she said ``Bill, you'll never be able to put it back together!!!'' I then ``tried'' to and ended up putting together instead a bathtub, a dog, a wall, and a W. But gee, it ``seemed'' like I was fumbling around and couldn't get a cube. ``Gee Bill, I think you've seen this puzzle before''. And who is this insightful girlfriend? My wife of over 20 years!)

Magic Magic Square (Sorry, dont know which talk) Try to construct a 4x4 magic square where (as usual) all rows and columns sum to the same thing. But also try to make all sets of four numbers that form a square (e.g., all four corners) also add to that number. Can you? If you insist on using naturals then I doubt it. Integers I also doubt it. But you CAN do it with rationals. How? If you want to figure it out yourself then DO NOT go to the answer which is at this link: here

Droste Effect: When a picture appears inside itself. For an example and why its called that see here

Black Hole Numbers: If you have a rule that takes numbers to numbers, are there numbers that ALL numbers eventually goto? If so, they are black hole numbers for that rule.

Map a number to the number of letters in its name

20 (twenty) --> 6 (six) --> 3 (three) --> 5 (five) --> four (4) --> four(4) --> ...

It turns out that ANY number eventually goes to 4.

Map a number to the sum of the digits of its divisors

12 has divisors 1,2,3,4,6,12 --> 1+2+3+4+6+1+2=19

19 has divisors 1,19 so --> 1+1+9 = 11

11 has divisors 1,11 so --> 1+1+1 = 3

3 has divisors 1,3 so --> 1+3=4

4 has divisors 1,2,4 so --> 1+2+4=7

7 has divisors 1,7 so --> 1+7=8

8 has divisors 1,2,4,8, --> 15

15 has divisors 1,3,5,15 --> 1+3+5+1+5 = 15

AH. It turns out ALL numbers eventually get to 15.

Boomerang Fractions: Given a fraction f do the following:

x1=1,  x2=1+f, x3- you can either add f to x2 or invert x2. Keep doing this. Your goal is to get back to 1 as soo nas possible.  Here is a paper on it: here. This notion can be generalized: given (s,f) start with s and try to get back so s. Can you always? how long would it take? Upper bounds?

Liar/Truth teller patterns on a square plane b Kotani Yoshiyuki. You have an 4 x 4 grid. Every grid point has a person. They all say ``I have exactly one liar adjacet (left, right, up, or down) to me.''
How many ways can this happen.  This can be massively generalized.

Speed Solving Rubit's cube by Van Grol and Rik. A robot can do it in 0.9 seconds: here.