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?