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.

Jan 25

• Looking for places where the landmarks line up in a given sequence;
• Guessed numbers in few questions
• Subtracted vectors...

Homework

1. Find places where you can see these landmarks in this orders (left-to-right): 

• Triangle-Star-Square-Circle
• Circle-Square-Star-Triangle
• Star-Triangle-Circle-Square

2. Draw the vectors A(1,3), B(-2,1), C(-1,-1). Find vectors A-C, A-B. 

Jan 18

Solve the trees between lampposts vs lampposts between trees puzzle. Can one see lamppost, tree, swing and climbing station in this orders: LTSC, CTLS? Other orders? All orders?

Find sums of vectors. Two vectors (2,1), (4,2). What vector added to the first gives the second?

Tile the plane with 4-gons: shares; skew squares; trapezoids; irregular 4-gons. Apple with 2 bites?

Five apples on the table; two are taken away - how many left? Five candles burning; two are extinguished - how many left?

Return to light bulbs puzzle...

dec 21: in lieu of math circle...

• Divide the square with two straight cuts into pieces so that each contains 2 cherries.
  
  
  
  
  
• Baby buffalo wants to go from her mom to her dad (visiting her aunt on the way. She is safe as long as she is closer to a buffalo than to any of the lions. How should she go?

• On a regular dice, the sum of dots on the opposite faces is always 7. 

 We cut the dice to get this development: can you reconstruct the dots on the left four faces?

december 7

Spent a lot of time rotating triangles.

Tried to get frogs to cross train tracks between two looooooong trains. The gap between them is long enough for a frog to get across, but how you move 1000 of them?

Tried to block a very quick cyclist from escaping: he is so fast that he rides in straight lines only. Here how one can block him with four rectangular cars:


Can you do it with three round cars?