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?

november 9

• More work with vectors. Can we detect a right angle? 
• Coloring maps: when two colors are enough?
• Getting robots out of the maze, blindly
• Choosing socks again. Walking in square grid?

Experimenting with Hopscotch robots...

homework

1.  How many colors one needs to color this map?

2. On the left is shown what would happen to the green strip under the turn by 90 degrees around the red point. Draw what would happen if you turn by 90 degrees the yellow and the violet shapes?

3. One big bag and one small bag can hold 5 pumpkins, while two big bags and three small bags can hold 12 pumpkins. How many pumpkins each bag can hold?

november 2

• More bus stop placement: this time, interactive. Also - on a circular road...
• Robot in a labyrinth - interactive.
• Choosing socks and walking in Urbana - same thing?
• What floats, and how...

Homework

1. If the drawer contains 2 red, 2 blue, 2 green, 2 yellow and 3 pink socks, how many one needs to take (not looking), to have at least 3 different colors? 4 different colors?

2. How many colors one needs to color
this map of an island, so that no two countries of the same color had a common segment of boundary?
















3. Black robot understands all the commands: UP, DOWN, LEFT, RIGHT correctly, but the white one does the opposite (you say RIGHT, it goes LEFT etc). They listen to your commands and move simultaneously. Can you make them meet,

if they start from these positions?

or from these?

october 26

• Continue choosing the position for a bus stop. How democracy works?
• Robot in a labyrinth
• Adding vectors
• Experiment: ball in a round glass

Homework:

1. Draw three vectors:

• A at (0, 0) 
• B at (0, 5) and
• C at (3, 4).

Which is further away from A: B or C?

2. Diza has a drawer with 4 green, 4 blue and 2 red socks.  She reaches inside, not looking, and grabs some socks. How many should she get, to be sure there are at least two socks of different colors?

3. Plastic container floats in a dish, so that the water level is an inch below the rim of the dish. If you move some of the water from the dish to the container, will the water level go up? down?

october 19

homework

• Where to place a bus stop so that the total walk for the people in the village is the least?

• Can you find all ways to choose two colors out of 6?

• It is easy to get a triangular shadow out of a tetrahedron. Can you get a square shadow?

october 12

Write numbers 1...9 so that any two neighbors were at most 2 apart and 9 is in the middle.

Diza has in her box red, blue and green socks. She never puts on socks of the same color. How many ways are there for her to choose her socks? If she has B, R, G and Y?

Envelopes for the cube

Envelopes for the tetrahedron...

Midpoints of vectors...

hopscotch! making the cupcake fly. Did not do today: vectors won.

Homework

1. In a box there are 5 pencils, Red, Blue, Green, Yellow and Orange. Joan wants to choose three of them. How ways are there to choose them?

2.

Mark on graph paper three vectors:

• A(3,3);
• B(-3,1)
• C(1,-3)

Draw the midpoints between A and B, B and C, C and A.

september 21

Partition lattice shapes into equal parts

Morse theory - try to move to abstract isoclines picture

Vectors

Bulls and Cows

Hopscotch: dino pursues the monkey getting bigger and bigger. Small steps, repeated...

august 24

Maxima and minima again

Magic squares - give an example; count the row/col sums; ask for another example

Hopping in a circle: every step, every other; every third... How long before coming back?

Adding vectors

Coding with Hopskotch

june 22

1. We'll cut squares - and more generally, rectangles, on the graph paper into 1x2 'domino's. 2x2 is easy. 2x3 too, and 4x4. Can one cut 3x3? What about 4x4 with 2 opposite corners removed?
2. More practice of finding remainders: when dividing by 2, by 3, by 4...
3. Back to graph paper: we'll be finding areas of various squares...
4. ...and adding vectors. This time, also trying to solve (1,2)+(x,y)=(3,4), or suchlike.

experiment: capillary forces, again, or perhaps, microwaving empty plastic bags...

june 15

1. We'll work again on residues, or remainders of dividing n by m: how many apples left if we try to split 7 between 2 people? 3 people?

Then, if we know that dividing between 3 leaves 2 apples, and dividing between 2 leaves none, how many apply one can have?

2. Some puzzles with lego pieces: how to assemble a solid 2x2x4 piece out of 8 1x1x2 pieces? How to assemble a blob with 5 colors so that each color touches each?

3. Can you design a baby version of SpotIt game? Say, 4 cards with 3 elements on each so that any 2 cards have exactly 1 element in common?

4. We'll try to learn how to add vectors on the plane (by concatenating programs that describe them).

For the experiment, we'll look at the capillary effect (microscope plates are handy!).

june 8

1. Some number of birds tries to sit on some number of branches.
If on each branch just one bird sits, one bird is left without a branch to sit. If the birds sit two a branch, one branch stays empty.
How many birds? How many branches?

What if:
again, if just one bird sits on each branch, one bird has no branch. If two birds sit on each branch, two branches are empty.
 How many birds? How many branches?

OK, one more:
If one bird sits on each branch, two birds are without a branch. If the birds two a branch, one branch is empty.
 How many birds? How many branches?

2. Anne, Boris, Cynthia, David, Ellis, Fatima and Greg sit in a row at a lunch table. When they opened their lunch boxes, it turned out that any two neighboring kids have exactly 3 apples between them. Anne has 1 apple. How many apples has Greg? How many apples they have altogether?

3. There are some kids in the room, having lunch. Any two of them have 3 whole apples between them. How many kids are in the room?

4. Two friends want to climb two paths - orange and purple - to the top of the mountain. They are friends, and want to stay all the time at the same height. Can they do it?

And what about these routes?

may 18

At the movies, Alice is left neighbor of Bob, Dorothy is right neighbor of  Carl. Dorothy and Alice are not neighbors. Who sits next to whom?

Quick Nick makes 4 laps, while Slow Joe makes just 1. If they start together, how many times Nick passes Joe? What if the numbers are 5 and 2? 5 and 3?

Draw boundaries on an island.

Vector addition.

Dragon, Unicorn and Sphinx again come to the restaurant.

may 11

Dragon, Unicorn and Sphinx again come to the restaurant.

At the movies, Alice is left neighbor of Bob, Dorothy is right neighbor of  Carl. Dorothy and Alice are not neighbors. who sits next to whom?

Quick Nick makes 5 laps, while Slow Joe makes just one. If they start together, how many times Nick passes Joe?

Four kingdoms on an island have all common boundaries. Can all four have also access to the ocean?

Vector addition.

april 27

A dragon eats for breakfast an apple and a boar. A unicorn eats for breakfast 2 apples. Some number of unicorns and dragons met for breakfast and ate three boars and seven apples. How many dragons and unicorns met?

Same question for five boars and eight apples. OK, nine apples. For which numbers this puzzle has an answer?

Pete goes from Champaign to Savoy (6 miles), making 2 miles per hour. Slow Joe goes from Savoy to Champaign, on the same road, making 1 mile per hour. How far from Champaign will they meet?

Graph paper and rulers. Coordinate axes. Place points. Are the three points on one line?

Straight line on graph paper: can it cross first two horizontal then two vertical bars? Write down sequences you see (VHHV, as an example).

april 20

Pascal triangle mod 2 (did not work, return later)

More Chinese reminders

Game with stickers on forehead: each reports how many blue/red stickers they see; define their own color. Two stickers?

Graph paper and rulers. Coordinate axes.
Place points. Are the three points on one line?

april 13

Add numbers separated by two. Is the result always even? Add three numbers separated by two... What now?

More Chinese reminders

Game with stickers on forehead: each reports how many blue/red stickers they see; define their own color. Two stickers?

Graph paper and rulers. Coordinate axes.
Place points. Are the three points on one line?

Digital straight line: draw line; see how the horizontal/vertical bars are intersected. Is there a pattern?

Unknot an unknot

copper tube experiment

4.6

Add neighboring numbers: why the result is always odd?
Add three neighboring numbers: what it is divisible by?
What about four neighboring numbers?

Pascal triangle mod 2

Post-it puzzle: Alice, Bob and Claire see the others' numbers, not theirs. Alice sees 3, Bob 3, Claire 2. What numbers each has?

Same with kids

Hedgehog wants to bring an apple to the porcupine, and return back

SET

Disentangle 

march 9

Chinese reminder theorem:

Some number of apples (stones) on the table (cover them).
If divided between 2 people, 1 is left, if between 3 people, none left.
If divided between 2 people, 1 is left, if between 3 people, also 2 left...
If divided between 2, none left, if between 3, 2 left. (8? what about 2?)

Right triangles. Define, look for right triangles. Split a given triangle into 2 right ones. 3? 4?

Carrying stuff around. Wolf-Goat-Cabbage problem.

SET

Round sticks balancing

February 16

Robots: wrote letter S (E4 N3 W4 N3 E4, or like); digit 7...

Navigated two robots in a maze past each other.

Word problems: Crocodiles got a package with fresh nasty boys. If each croc tries to get 2 boys for lunch, one croc will get none. If each croc gets just one boy, there will be one boy left. How many crocs and boys were there?

January 26

Program robots to write words.

Symmetry again - this time with several axes

Goat problem: where the grass was eaten?

Gallery guards: how may ways to place them?

SET!