November 17

• Warmup: 
• what is bigger, $754/344$ or $75/34$? 
• tile the plane with devils
• Binomial coefficients. Pascal triangle.
• Return to counting paths in a graph...
• Sets. Formulae. Subsets: predicates and interpretations.
• Game B&C?

November 10

• Warmup: 
• what is bigger, $354/734$ or $35/73$? 
• what is the last digit of $1\times 2\times \ldots\times 11\times 12$? 
• Induction: cut 2^n x 2^n square - one corner into L-trigons.
• Squares on the surface of $10\times 10\times 10$ cube are painted in white and black.
• How many squares are altogether?
• If 301 square is painted white, is there always a pair of opposite white squares?
• Return to counting paths in a graph...
• Sets. Formulae. Subsets: predicates and interpretations.
• Game B&C?
  

Homework

• Draw the areas described by the formulas

November 3

Past homework:

• prism+tetrahedron
• cutting the cube to get regular 5-gon

New:

• Extending faces of cube, they partition space into how many pieces?
• Same about tetrahedron, prism, octahedron
• x^3=5/4 y^2=9/8. What is greater, x or y?
• Roll coins
• Wire-frame with sides 3,4,12. That is the distance between endpoints?
• Induction: cut 2^n x 2^n square - one corner into L-trigons.

Sets,

October 27

• Warmup:
• $x^5=2; y^8=3$. Which is greater, $x$ or $y$?
• Turning coins around each other
• Ball jumping, first jump 1/2 foot; each next jump 20% shorter. How far will it go?
• Pythagorean theorem. 
• How to find the diagonal of a brick?
• Back to residues: build tables for mod $5,6,7,8$. 
• No zeroes in the tables for what modules?
• What happens if we multiply rows?
• Game - bulls and cows...

October 20

• Homework: pentahedron; origami?
• If $x^3=26$ and $y^2=9$, what is bigger, $x$ or $y$? 
• Two players try to move short hand to 5 o'clock. They start at 1, and can move it ahead 2 or 3 hours. Who wins?
• Geometric progressions - how to find their sums.  When the sum converges. 
• What about other convergences: example of $1+1/2+1/3+\ldots$
• $a^6\mod 7$ for $a=1,2,\ldots,6$.
• The following sets are given:$$A = \{1, 3, 7, 137\}, B = \{3, 7, 100\}, C = \{0, 1, 3, 100\}, D = \{0, 7, 100, 333\}. $$
  Describe the sets:
• \(A \cup B\);
• \(A\cap B\);
• \((A \cap B)\cup D\);
• \(C \cap (D\cap B)\);
• \((A\cup B)\cap (C \cup D)\);
• \((A\cap B)\cup (C \cap D)\);
• \((D\cup A)\cap (C \cup B)\);
• \((A\cap (B\cap C))\cap D\);
• \((A\cup (B\cap C))\cap D\);
• \((C \cap A)\cup ((A\cup (C \cap D))\cap B)\)

October 13

•  Problems with remainders
• $39^{192}\mod 11$;
• $101^{1001}\mod 7$;
• $a^6\mod 7$ for $a=1,2,\ldots,6\$.
• Tetrahedra and pyramids.
• The skeleton of a tetrahedron made out of wire. Play the game: cut an edge, in turns, until it falls apart. Who wins?
• Same for pyramid with square base.
• To a triangular face of a regular (solid) square pyramid one glues on the regular solid tetrahedron. How many faces the resulting solid body has?
• More experiments with Polish notation:
• $+++++++1,2,3,4,5,6,7,8$.
• Does the answer depend on the order of numbers?
• What about $+1+2+3+4+5+6+7,8$?
• One can get different trees - can one draw all of them?
• The following sets are given:$$A = \{1, 3, 7, 137\}, B = \{3, 7, 100\}, C = \{0, 1, 3, 100\}, D = \{0, 7, 100, 333\}. $$
  Describe the sets:
• \(A \cup B\);
• \(A\cap B\);
• \((A \cap B)\cup D\);
• \(C \cap (D\cap B)\);
• \((A\cup B)\cap (C \cup D)\);
• \((A\cap B)\cup (C \cap D)\);
• \((D\cup A)\cap (C \cup B)\);
• \((A\cap (B\cap C))\cap D\);
• \((A\cup (B\cap C))\cap D\);
• \((C \cap A)\cup ((A\cup (C \cap D))\cap B)\)
• Origami!
• Game: bulls and cows

October 6

• Continuing with the algebra of remainders. $15^{23}\mod 7=?$
• Create your own problem!
• Recall how to solve linear equations,
• and use it to find sums of geometric progressions. 
• ...like $1+2+4+8+\ldots=-1$!
• How to write down arithmetic expressions:
  $$(2+3)*5-7$$
• ...using trees!
• or Polish notation
  $$-*+2,3,5,7$$
• Trees are useful in syntactic ambiguities:
  The professor said on Monday he would give an exam...
  One morning, I caught an elephant in my pajamas...
  The professor attacked the student with the umbrella...
• Back to sets: complements. What is $\overline{A\cup B}$?
• Game: bulls and cows

September 22

• Still trying to draw a bicycle!
• Pack squares with side \(1/2, 1/3, 1/4,...\) into the square with side \(1\).
• Find products of some triples of consecutive numbers. Are they divisible by \(6\)? Is this always so?
• What last digit of a product depends on? Find last digits of some pairs, like \(23\times 56\) and \(53\times 36\). Are they always the same? What is the last digit of \(847281937\times 938274\times 9629052717\)?
• How the remainder of division of \(n\) by \(10\) is related to the last digit of \(n\)? What is \(n\) is negative?
• Find the last digit of \(17^{17}\).
• In general, what happens with the remainders \(\mod p\) for some \(p\), like \(5\) or \(9\), when numbers are multiplied? 
• Sets: intersections (AND) and unions (OR). 

Homework:

• If one finds the sum of digits of \(17^{17}\), then sum of its digits, and so on, till one digit remains, - what is it?
• The following sets are given:\[A = \{1, 3, 7, 137\}, B = \{3, 7, 100\}, C = \{0, 1, 3, 100\}, D = \{0, 7, 100, 333\}. \]
  Describe the sets:
• \(A \cup B\);
• \(A\cap B\);
• \((A \cap B)\cup D\);
• \(C \cap (D\cap B)\);
• \((A\cup B)\cap (C \cup D)\);
• \((A\cap B)\cup (C \cap D)\);
• \((D\cup A)\cap (C \cup B)\);
• \((A\cap (B\cap C))\cap D\);
• \((A\cup (B\cap C))\cap D\);
• \((C \cap A)\cup ((A\cup (C \cap D))\cap B)\)

Reminder: no class on Sept. 29.

September 15

1.  Can you draw a bicycle? Answer: unlikely!
2. 8 rows, 7 seats in each; how many tickets (randomly taken) ensure that you have two seats next to each other in the same row? Answer: 33.
3. Is it possible to cut pizza into 5 pieces so that after everyone eats the cheesy part, 6 pieces of crust are left (no crust breaking)? Answer: Yes! What's the largest number of pieces of crust can be left? Answer: 8.
4. Sets: empty sets, subsets.

Game: Rock-Scissors-Paper, played collectively (whoever beats the biggest team, wins).

Homework:

1. The following sets are given:
   \[A = \{1, 3, 7, 137\}, B = \{3, 7\}, C = \{0, 1, 3\}. \]
1. Which of them are subsets of some other set?
2. How many subsets each of them has?
2. Among the numbers \(1/2, 1/3, 1/4, 1/5, 1/6, 1/7 \ldots\) choose 5 so that their sum is equal to 1.

May 5

• Warmup: 
• Devil offers another deal: each time you clap, he squares your amount of money, but then takes 99% away. How much you need to start with to actually profit from the deal?
• Find the remainder of 2222.....222/7 (1000 2's)
• Binary numbers: review what we know; algorithm of finding binary representations.
• Convert 2033 and 10001 into binary. Convert 1011110001 and 10000111101 into decimal.
• Adding numbers. Subtracting, multiplying...
• Binary numbers are longer than decimal, - by how much?

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: