may 24

Can one tile the whole plane with 2 monominos, 2 dominos, 2 triminos etc (all - linear).

Recall Nim's; move to Nims with 2 piles; Nims with 2 ends...

Intermission: CCXLVIII-CXXII=?

Split payment fairly...

Homework

may 17

• which of these tiles can tile the whole plane?
  


• Analyze Nim₄, Nim₅. What about Nim_1,3 (you can take 1 stone or 3 stones, but not 2...)?


• Can you partition with one straight cut these figures so that each part has the same area of red and white?
  
  
  
• Split payment fairly...

Homework

• Can one tile the plane using these shapes?
  
  


• Create the table of winning and losing in the Nim_1,4 game (one can take 1 or 4 stones). Who wins if you start with 99 stones? 100? 101?


• Cutting the shapes! - the problem above is still unsolved!

may 10

• Nim (or Nim₂, where one is allowed to take away 1 or 2 stones, other Nims are possible) - the complete solution is easy: if one faces the pile with the number of stones there divisible by three, one loses given the opponent plays wisely, and wins otherwise. We established that using the recursion, an awfully useful trick.
• This led to the question, how you determine divisibility by three? The sum of digits trick helps...
• Draw projections of some shapes
• Split into 3 parts led to the question: can one tile the plane with these pentaminos?

Homework:

• One can tile (cover without overlaps or holes) the whole plane with the r pentaminos, for example as shown below:
  
  Now, can you tile the whole plane with T-pentaminos? Or with y's? or z's?
  

• Is 198407392 divisible by 3? What about 38002978643001?
• Who wins the game of Nim₃ starting with the pile of 101 stones?

may 3

• We finish the three different coins puzzle (how to order them with fewest weighings on a scale)
• We'll work on the Nim game: what is the best possible strategy? Who wins if one starts with 100 stones?
• Vectors! we will add more than one vector; and see what kind of configurations sum up to zero...
• We play more of the "pick a symbol" game...
• Time permitting, we'll talk about splitting the payment for rice bowl...

Homework!

• We established the pattern of whether the person wins or loses facing a pile in the Nim game - one wins if the pile has 1 or 2 or 4 or 5 stones; loses with 3 or 6... 

  1	2	3	4	5	6	7	...	...	99	100
  W	W	L	W	W	L	W	...	...	?	?

  Can you say what happens when you have a pile of 99 stones? of 100?
• Roman numerals strike again: Find CIX+LXXI-XL=?
• Split into three equal shapes:
  

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)).