april 26

• We started with yet another weighing problem: one has 4 coins that look identical, but two of them are heavier than the other couple. How many weighings on a scale we need to detect, which two are heavier?
  Answer: 2 weighings is enough.
• We added numbers: CXC+XIX=?
• The game of Nim is simple: two players take in turn from a pile one or two tokens from the top. Whoever takes the bottom token, wins.
  We established that in piles of size 1 or 2, the first player wins, in the pie of 3, the second player does - provided they make the best possible moves...
• Lastly, we played an unusual game: one had to choose (secretly) one of eight objects, and the voters for the most popular one would be the winners.
  Then we changed the rules: the voters for the least popular would be the winners.
  Then - the voters for the object getting exactly 2 votes... 

Homework:

• Now, you have 3 coins, all of different weights. How many weighings you need to find the heaviest, the lightest coins?
• In the game of Nim, who wins if there are 6 tokens in the pile?
• Add CXL+XXXVII+XXIII=?

April 12

• Several games:
• On a city map, take a point: if you meet most of your friends, you win.
• Think of a number (1-9) - those with most claimed number, win.
• Those with least claimed number win.
• Sharing stuff: Pete and Claire have 2 and 4 cups of rice. They cook together a wok of delicious fried rice. 
• A stranger comes to them, and buys a third f their rice, paying $5. How they have to share the money?
• What if they had in the beginning 2 and 3 cups - how they should split the money now?
• We will do vectors (from the homework and like)
• Roman numerals, again! what is XC + XXIV?
• We'll continue statistical estimation experiments...

April 5

Homework:

• Recall, that if you have 9 lookalike coins, one of which is fake (heavier), you can find it  with just 2 weighing on the scale (remember how?).
  Can you do the same with 10 coins? Can you find the fake coin among 10 coins with 3 weighings? among 11 coins with 3 weighings? 20? 25? How?


• If you lost you friend in Paris (and neither of you have a cell phone!), how you would look for her?


• If you have  vectors A(1,2), B(-2,1) and C(1,-3), what is A+B+C? (Recall, that you have to walk these three vectors one after another, starting at (0,0)).

march 1

1. We talked more about the Roman numerals. Converting them into the usual (arabic) numerals feels easy; but writing numbers using them - not too fast!
   
   
2. We worked more on the restaurant problems: this time Rabbits (2 Cabbages) and Piglets (1 Cabbage and 1 Apple) were visiting. How many R's and P's are there if they ate 6 Cabbages and 4 apples? 8 Cabbages and 2 Apples?
   
   
3. We started to work on the jug problems, like this one:
   
   
   We'll return to this problem next Sunday...
   

Homework 

1.  Write these numbers as Roman numerals: 40, 48, 140, 147, 80, 90, 98.
2. How many Rabbits and Piglets were there is 10 Cabbages and 6 Apples were eaten?
3. You have two jugs, of volumes 5 and 8 liters. Can you measure precisely 2 liters?

feb 22

1. We found the number of ways a slalom skier can go through the following courses:
   
   
   
   Here are the answers - 4, 6 and 8 ways:
   
   
   
2. We looked at the old puzzle in a new way. In a cafe frequented by dragons and unicorns, a bill was showing 3 cabbage heads and 3 apples. Recall that unicorns always eat 2 apples and a cabbage head, and dragons always take 2 cabbage heads and 1 apple each. How may dragons were there? The idea is draw a vector plane, with the horizontal steps representing cabbage heads, and vertical - apples. Each dragon eats a vector of (2,1) (two cabbages, one apple), and each unicorn eats (1,2). So, to get to (3,3) - three cabbages, three apples, shown as a purple point - one needs to make one "dragon step" and one "unicorn step":
   
3. We learned about the Roman numerals: IV is 4, but VI is 6... XLIX is 49!
   
   To remember: I is 1, V is 5, X is 10, L is 50, C is 100.

Homework:

1. How many ways are there to go down this course:
   



2. On the graph above, how many dragons and how many unicorns were eating at the cafe, if the bill was represented by the orange vector (5 cabbage heads, 7 apples)? yellow vector (8 cabbage heads, 7 apples)?
   
   
3. What are the numbers written in Roman numerals as XXIX? XIV? XXXIII?

Feb 8

• Do the cut (via squeeze)
• Cube
• Circular road; seeing permutations: reversed if seen from opposite ends. Not always so!
• Two hens lay two eggs every two days. How many eggs six hens will lay after six days?
• Vectors...

Homework:

1. Four legos are hiding under a sheet of paper:

Here is what you see from the left (green arrow):

and here is what you see from the right (red arrow):

Can you determine how there are positioned under the sheet?

2. Two of these links are the same, and one is different. Which one? (You cannot cut the loops!)

Feb 1

Not too late for the homework!

• Four landmarks, and a circular road going around them. In what order do you see these landmarks as you walk along the road?
  
  
  
• Still unsolved! how to cut this with two straight cuts into four pieces with two red dots in each?
  
• Draws vectors A(2,2), B(-3,3), C(0,-1), and find the vectors B-A and A-C.