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?