April 28

• Warmup: 
• Devil offers a deal: each time you clap, your amount of money doubles, - but you'll pay me $64. You agree, but after 3 claps all you had is gone! How much you started with? How much you need to start to grow richer and richer?
• Find a number consisting only of 1 and 0's which is divisible by 3. By 9. 
• What about numbers made of 8 and 5's?
• What about divisibility by 7? 1001. Remainders.
• Binary numbers: weights 1,2,4,8 etc: represent 1, 5, 14, 17... Adding numbers. Subtracting.
• More shear transformations. What transformations preserve shape?

April 21

• Warmup: 
• Adding 0 makes 2-digit number 252 bigger, - find the number
• Adding 0 between 2 digits makes it 9 times bigger
• Moving 9 from 1st to last (3rd) place increases the number by 216
• Number trick:
  x-> 3x-> 30x-> 30x-5-> 6x-1->6x+12->x+2->2 
• Design your own trick
• Coordinate system: shears
  S:(x,y)->(x+y,y) 
• Apply to squares and rectangles.
• U:(x,y)->(x,y-x).
• Find SU. Find SU^6.
• Game: card guessing?

April 14

• Warmup: 
• Homework: discuss oil/vinegar problem; moving candy into one place... 
• Three boxes: first has 6 less candies than 2nd and 3rd together; second has less candies than 1st and 3rd together. How many candies in the third box?
• Cop runs 3 times faster than a robber. Can cop spot the robber in a city of 4 square blocks (3 streets and 3 avenues)? 
• Periodic patterns: each 2x2 square on an infinite grid contain U D R C (unicorn, dragon, rabbit, cat). Is it true that each row or column contains all critters?
• Linear transformations: we know symmetries; what about shears?
  S:(x,y)->(x+y,y)Find images of a few points. Draw what happens to a cat.
• Do S^2. Find S^-1.

April 7

• Voting schemes: 8 R's vs 19 D's still can elect their man, in three-tiered elections.
• Cutting into equal parts
• Invariants: turning 3x3 table into all 0's starting w one 1 by flipping rows or columns: impossible!

Homework:

• Take 1 pint of oil (in the cup A), 1 pint of vinegar (in cup B); put one TBS of oil into cup B, shake a bit; take 1 TBS of the mixture and put it back into the cup A. Now you have two mixtures: what has more, vinegar in the cup A or oil in the cup B?
• Apply Ax, Ay and AxAy to this dog (recall: Ax is the reflection in the x axis; Ay - in the y axis):
  

Feb 10

Warmup problems:

• In an analog clock, how often small and big hands overlap during 24 hours?
• …how often they are opposite?
• …at 90 degrees?

1/13=0.0769(53841)...  periodic decimal fraction.

3/17=0.1765... - takes a while, but still decimal expansion is periodic.

We (sort of) proved: a number is rational if and only if it has periodic decimal expansion.

Areas of parallelograms: determinant: for a pair of vectors (a,b) and (c,d), the area of the parallelogram spanned by them is ad-bc (this expression is called determinant).

Sometimes long vectors generate parallelograms of small area, for example (13,21) and (21,34)... (Hmm.... Fibonacci numbers here for some reason...). Sometimes the area of the paralellogram is just 0! (For example, for (2,2), (5,5))

Game: each  player starts with $6, contribute any amount to the general fund. Then the general fund is distributed equally among with exception of those who contributed least.

 Homework

• 0.12345678910111213141516... (we just write down all the natural numbers in their decimal form). Is this number rational?
• 22/23=0.95652... Can you find period here? 
• Find a lot of pairs of vectors with zero area.
  
  

February 4

How to turn periodic decimal fractions into usual fractions...
For example: 0.303030303... Call this number x. Then 100x=30.3030303....=30+x. So 99x=30, and x=30/99=10/33.

How to find areas of parallelograms...

And the game was to disentangle yourself.

Homework:

1. Write 0.270270270270... as a fraction.

2. Find the area of parallelogram in the picture:

3. Who is linked with whom:

Dec. 2

Recall that {x} is the fractional part of a number, that is how far it is from the larger integer less or equal to x.

Exercises:

• Try to compare in your head 192/133 and 79/53.
• If {x}=1/2, and {y}=1/2, what are possible options for {xy}?
• What are possible options for {x}+{-x}?