Friday, October 10, 2014

Fold me to the moon

On September 27, Dr Emily Evans led our Junior Math Circle, with the following activity.

As a warm up activity / pre-activity, each table was given several sheets of square paper and instructions for a simple origami dog and instructions for an origami crane.  The students worked on folding these until we had enough students to start.

Since it was close to the beginning of the year we spent some time talking about the four rules of math circles:

1. Have Fun
2. Ask Questions
3. Make Mistakes
4. Help others have fun.

Each student was given a fresh piece of paper and their adult helper was given a chart to fill out.  The students were instructed to fold the paper in half, then unfold the paper and count the number of rectangles.  The adult helper recorded the number of rectangles in a chart. The students then refolded the paper and then folded it in half again.  Some students quickly saw a pattern, number of rectangles to number of folds, but they were encouraged to continue folding and counting to get some information for the next part of the activity.

After most of the students were done folding we talked about the pattern that we saw.  We then filled out a table on the board up to 10 folds.  On the board we graphed the number of rectangles vs. the number of folds for up to 5 folds.  I told the students that mathematicians call this an exponential curve.  The students could see how when you double things they grow very quickly.

How quickly do things grow?  We observed that every time the paper was folded it got thicker.  The students were then asked how many times they thought they would have to fold the paper until it was as tall as the Y on the mountain (you could substitute the Statue of Liberty since it is essentially the same height).  We then did a doubling exercise.  I told them a piece of paper was .0001 meters thick and the Y was 94 meters tall.  We then made a chart on the board with how thick the paper was after each fold.  After 5 folds the paper is .0032 meters thick, After 10 folds the paper is .1024 meters thick or about four inches.  After 15 folds the paper is 3.277 meters thick or about as tall as the ceiling.  After 20 folds it is over 104 meters tall, or higher than the "Y" on the mountain.  (We went through the first twenty folds on the board).  Since the students were restless at this point I just mentioned that if the paper was folded 42 times it would reach to the moon.

We then talked about how many times they actually were able to fold the paper.  The responses were 5-7.  We talked about how they could fold the paper more times.  We came up with two ideas, bigger paper and thinner paper.  The students were given either very thin tracing paper or very large sheets of butcher paper and given the opportunity to try again.  They got the same answers as before.  We then tried to identify paper that was both thin and large... the answer toilet paper.

The students then broke into smaller groups and were given a roll of toilet paper and asked to fold it as many times as they could.  This took a while.  Most groups folded the paper about 10-11 times.  We talked about how many rectangles would be on a single piece of toilet paper.

Then it was time for cookies and to depart until the next week.

Passing Candy

I was away for the last half year, and stopped updating this website. I'm back now, and our junior math circle has already run a couple of times this year, so I'll try to be better and keeping our activities posted.

Two weeks ago we held our first Math Circle of the year. Again we invited children in kindergarten through 3rd grade -- older students could go to other classes, and requested that the young children bring a parent. We had about 20 children show up, and about 10 parents.

Since it was our first session, we went over some rules of Math Circles:

1. Have fun.
2. Help others have fun.
3. Ask questions.
4. Make mistakes.

(One little girl also suggested "listen to the teacher," but I like to stick with the four basics above and let the parents take care of having their children listen.)

The activity of the day, for our first session, involved candy! It was adapted from an activity due to James Tanton, from his book "Solve This."

The children were sitting at round tables, each table with three to six children. I told the children they would be playing a game with candy. I would give each of them some candy, and their job would be to divide their candy exactly in half, to keep half, and pass the other half to the person sitting to their right.

Before I handed out candy, we talked about what it meant to divide something in half. I asked when it would be hard to divide a stack of candy exactly in half, and the children told me that it would be hard if they had an odd number (although you could cut a piece of candy, or bite it, they suggested). We had a little discussion about even and odd numbers.

If you have an even number of candy pieces, like 2, 4, 6, 8, 10, 12 pieces, then you can divide your candy exactly in half.

If you have an odd number of candy pieces, like 1, 3, 5, 7, 9, 11, then you can't divide it in half without cutting it.

So I added a rule: if you have an odd number of candy pieces during our game, raise your hand and I will bring you one more piece, so you then have an even number.

I then passed out exactly 2 pieces of candy to every child in the room. When they were ready, I asked them to divide their pile in half, and keep half and pass the other half to the person on their right. We did this one time, and I asked what had happened. Had anything interesting happened?

Everyone noticed that they all still had 2 pieces of candy. We talked for a minute then about why, and about what would happen if we kept playing the game forever and ever. They all could see that everyone would always have 2 pieces of candy. Then I gave them a challenge. At your table, take the pieces of candy I have given you, and distribute them in a different way so that you can get the most candy from me that is possible! That is, try to make odd numbers, so you can raise your hand, so I will come around and give you more candy.

The children were pretty excited about this game. Several tables had one child pass one piece of candy to the person sitting next to them, so around the table the numbers of candy were 1, 3, 1, 3. Other tables (with an odd number of children) came up with different strategies. When they were ready, I had them divide their stack in half and pass it along. Those with an odd number raised their hands, and I hurried around the room giving extra candy to the students who needed it.

We played several rounds. After a while, at every single table, every child had exactly four pieces of candy. They realized that no matter how many times they kept playing, once everyone had four pieces, everyone would keep exactly four pieces of candy forever.

At this point, I suggested we change the rules. Rather than raise your hand when you had an odd number of candy, eat a piece of candy instead! Again, I had them start over, trying to distribute the candy so they could eat as much as possible!

We started again and played several rounds. After a little while, every table was in the situation that each child had exactly 2 pieces of candy. They realized they wouldn't be eating any more candy.

Now, at this point, I asked the children if they thought they would always end up with the same amount of candy. They weren't sure, so I suggested they try again, with the game of adding candy rather than eating it, and see if they could get a different result by starting by distributing the candy differently. I wandered the room while they tried.

Playing this game basically used up the rest of the time. None of the children were able to find a way to get more candy. I challenged the tables to see if they could figure out why, but we didn't really have the time to pursue this more carefully. (Maybe I'll bring it back for some other activity -- the answer isn't too tricky I think, even for children this age.)

Although we didn't solve all the problems, at the very least the students left knowing how to split small integers in half, how to tell the difference between even and odd, and with candy stuck in their teeth on an early Saturday morning!

We ended with refreshments. (Not candy.)

For further thought: No matter how the children distribute their candies, they will always reach a stable point where all the children have the same even number of candy. Can you figure out why? 

Saturday, November 16, 2013


Dr Pace Nielsen led today's math circle. The topic was dominoes. A domino is a rectangle made of two squares (see below).

Dr Nielsen:  How many different ways are there to arrange dominoes?
The students came up with four ways, shown below.

Rules:  No leaning a domino on its side, turning it upside down, cutting it in half.

Dr Nielsen:  Today's problems have levels!

Level 1.

Question: Can you fill up a 1x1 chessboard with dominoes?

Children:  No!  Because a domino doesn't fit!

Level 2.
Question:  Can you fill up a 2x2 chessboard with dominoes?

How many think you can? (Many hands).

OK you have a piece of paper and a pencil. Try it.

After a minute or two, a girl volunteered to show how to cover a 2x2 chessboard with dominoes.

Notation: draw a line through two squares to represent a domino.

Level 3.  Can you fill a 3x3 chessboard with dominoes?

The children worked for a minute.  After a while, several raised their hands to say no.

Why not?

Because dominoes have two squares each, so they cover an even number of squares.  But there are nine squares in a 3x3 chessboard, which is odd.

Level 4.  4x4.  Can you do it?

Again after a couple of minutes, a student came up and showed one solution on the board.

Dr Nielsen:  Have you figured out a pattern?

The students decided that you could always fill a chessboard with an even number of squares, but you could never fill a chessboard with an odd number of squares.

Level II.

Dr Nielsen:  Can you cover a 2x2 chessboard with dominoes if one of the corners of the board is cut out?

The children thought for a minute, and then answered no.

One child had a suggestion:  You can do it if you let the dominoes overlap.

Dr Nielsen:  Very good!  New rule.  No overlapping dominoes.

Then the children could see that there were an odd number of squares, so they couldn't cover this board with dominoes.

Next level:  What about a 3x3 board with one corner missing?

One child figured out a solution and showed it on the board.

Next level:  What about 4x4 board with a corner missing?

Immediate answer. NO!

Why not?

Because there are an odd number of squares.

One of the children who knew how to multiply explained that an even number times an even number is even, and if you take away one, you'll get odd.

Those children who didn't know how to multiply could just count:  15 squares.  Odd, so you can't cover it with dominoes.

Next level:  What about a 5x5 board with a corner missing?

 After a minute, one child finished and raised his hand.  Dr Nielsen asked that child to try the 7x7 board while the others finished.

After a few more minutes, when many hands were raised, Dr Nielsen had the children give him the answer.

Can you cover a 5x5 board with a corner missing?  YES!

Can you do 6x6?  (Immediate answer)  NO!

What’s the pattern?

The children decided you could always do odd sided squares with a corner missing (3x3, 5x5, 7x7), but you could never do even sided squares with a corner missing (2x2, 4x4, 6x6).


Level 1.  Can you fill a 3x3 chessboard with dominoes when there are two opposite corners missing?

Children:  No.  Because there are seven squares left over.  (They seemed to be figuring out a way to solve these problems!)

Level 2.  What about a 4x4 board with opposite corners missing?

One child:  Yes!  I just need to figure it out.

Dr Nielsen:  Ok.  Figure it out.

After a few minutes, the children changed their minds.  No!  You can't do it!

Dr Nielsen:  Are you sure?

After a couple more minutes, with still no children who had found a solution, Dr Nielsen asked, How many squares are left?

Children:  14.

Dr Nielsen:  14 is even.  It’s even!  So you can do it right?

Children:  No.

Dr Nielsen:  Why not?

One child suggested that the board had been "cut up too much."

So Dr Nielsen tried:
 Dr Nielsen:  Well, that didn't work.  But maybe I just chose poorly.  Maybe I can do it if I'm more careful.

(The children were skeptical, but without a good answer as to why not, he moved on.)
Dr Nielsen:  We know we can’t do a 5x5 chessboard, because when we remove 2 corners there will be 23 squares, which is odd.  Let’s try 6x6. 

The children worked on it for several minutes, with Dr Nielsen asking every now and then how many people needed more time?  Since no one had quite finished, he let them keep working.

After a while, he brought the group together to talk about the chessboard.  

Dr Nielsen:  Let's color the 6x6 chessboard black and white.

How many squares are there total?  36
How many black squares?  18
How many white squares?  18

If we remove the top left corner and the bottom right corner, how many black squares and how many white?

18 black, 16 white!

What does a domino cover?  1 black square, and one white square.

So can you cover a 6x6 chessboard with opposite corners removed?

No!  There aren't enough white squares!

A student noticed that for an odd number, like a 5x5 chessboard, opposite corners had different colors.  They asked, could you cover a 5x5 chessboard with opposite squares removed?

Dr Nielsen:  Good question.  Let's try it!

After a minute, the students realized that they had already figured this problem out -- there were an odd number of squares, so no, they couldn't cover this chessboard with dominoes.

One last thing.

With the remaining time (not much of it), Dr Nielsen had the students create an addition table on the board, adding numbers from 1 to 5.

After they had finished the table, he pointed out that they could make an addition table for even and odd numbers.

And for those who knew multiplication, they could do a multiplication table for even and odd numbers.

And then we broke for cookies.

Saturday, November 9, 2013

Checkerboard problems

For today's math circle, I adapted a couple of activities from James Tanton's book Solve This: Math Activities for Students and Clubs. 

The above book, by the way, is a great book that describes activities that Tanton used while running a math club for college-aged students.  Unfortunately, my students are aged 5 through 8, so not everything in his book will work in our class.  Some of the activities are a little too mathematically sophisticated for children who can't yet multiply.  And other activities require a bit more attention span than that of your typical kindergartener.  But all that said, in fact a surprising number of the activities in the book work even for this age group, especially with some minor modification.  Today's activity was one of those. 

Specifically, we were looking at problems adapted from Chapter 13 of the book.  If you're following along with your own copy of Tanton's book, you'll see that to the activities he gave, I added a couple of easier cases to work through first (to warm up the younger children), and I simplified one activity slightly (reduced from a 5x5 grid to a 3x3 grid).  Also, I stuck with just two activities rather than try to do all three or more activities listed in there.  But otherwise, my activities were pretty similar to his.

1.  Prepare one copy of "easy" grid puzzles below for each child, and at least three copies of the "hard" grid puzzles for each child. 
2.  Bring enough pencils for each child.
3.  Just before class, I used masking tape to mark nine x's on the floor for the people-shuffling activity (2nd activity below).

Grid Puzzles.

I learned from last week that any activity I start right at 9:00am (when our math circle starts) will have to have its rules repeated as kids come a little late.  So today, I started with an activity with paper and pencil that was easy to explain to late arrivals.

As the children arrived, I handed out pencils and a paper with the following squares printed on them:

The rules of the game are the following.  Start with your pencil in the square marked with the X.  Draw a path through the grid that meets each square in the grid exactly one time.  The path may leave a square through any of its sides, but it can't run diagonally out of a square.  (Examples are below if that explanation doesn't make sense.)

Most of the children figured out paths that worked for the above four puzzles quickly, especially the older ones.  As the children finished, I handed them a harder puzzle.  Here is what the harder puzzle looked like. 

Puzzle #5 was difficult, but the students were having particular trouble with puzzle #6.  I brought several extra copies of the puzzle so that when they had erased too many times, they could get a fresh puzzle and try again.

I let them work on this for about 15 minutes.  Then I polled them to see who had finished their puzzles.

Everyone had finished puzzles #1 through 4.  I let the children raise their hands and describe different solutions to puzzle #1.  Here are three that they came up with.

I asked for a show of hands on who had finished puzzle #6.  No one had.  Same for #7.  ("We didn't get to #7 because we couldn't finish #6!" explained one child.)

Rather than talk more about these puzzles, I told the children we were going to take a break from the puzzles and do the next activity.

People shuffling.

I asked for four volunteers to come play the next game.  They stood on X's marked with masking tape on the floor in a 2x2 grid. 

The rules of the game are the following.  Every student must move exactly one space.  They can move side to side, but not diagonally.  The goal is to get every person to move to a different space in the 2x2 grid.

I had my four volunteers run through an example.  They switched places in pairs, which worked!  I then asked all the students to get into groups of four and see if they could figure out different ways of solving the problem. 

We took about five minutes, then put some solutions on the board. 
The children could switch places in pairs in two different ways, or they could move in a cycle of four two different ways. 

I then asked for nine new volunteers, and asked the children to stand on a 3x3 grid.  The rules were the same, the objective the same:  Everyone has to move exactly once, with no diagonal moving allowed. Ready set go.

Their first attempt didn't work -- someone on the corner got stuck. 

One eager and clever little girl had an idea then.  Have the middle person move first, then everyone else switch around them.  But unfortunately, that didn't work either.

Another equally clever girl suggested a new alternative.  But hers didn't work either!

I then suggested we try moving one person at a time, counting how many moves were made before someone stepped back into an empty space.  A sequence of legal moves of people that ended with someone taking the empty space was called a cycle.  We worked through a few cycles, and found that they all had to have an even number of steps.

Why was the number even?

This is where things got a little tricky for the younger kids.  I showed them that every time someone moved left in a cycle, someone else had to move right.  Every time someone moved up, someone else had to move down.  That meant moves in the cycle happened in pairs -- so there were an even number of moves!

A couple of the older children seemed to get it now.  Because there were nine children, but cycles had an even length, the only way to get everyone to move would involve an even number of children.  So one of the nine would be left out.

(I don't know if they really got it, but at that point, the littler ones were getting restless, so I had them all sit down again.)

I had a couple other moving puzzles prepared, but the students voted to go back to the grid puzzles.

Grid Puzzle Solutions.

One of the girls who had been helping with the 3x3 people-shuffling game raised her hand and said she thought that solving puzzle #6 (grid puzzle above) was impossible, just like moving nine people around in a 3x3 grid was impossible.

I announced to the class that she was right!  Puzzle #6 was impossible.  And our new goal was to figure out why. 

Someone suggested that maybe it was because there were 25 squares -- an odd number.  But another child pointed out that puzzles #4 and #5 also had an odd number of squares, but we were able to solve them.

Then a boy noticed that in puzzle #6, there were only 3 ways to begin, but in puzzle #5 there were 4 choices for how to begin.  That was a good idea.  But then someone realized that in puzzle #4, there were only 2 choices for how to begin, but everyone had solved puzzle #4. 

I told them I would give them a hint.  I drew the 5x5 grid on the board, and started coloring the squares in a checkerboard pattern. 

I then let them think for a while and talk about the problem with the others at their table. 

They noticed that #4 and #5, which were solvable, started on shaded squares.  But #6 and #7, which were not, started on white squares. 

Was starting on white squares the problem?  Maybe, but puzzle #3 also started on a white square in the 4x4 grid. 

By then the children realized the problem was with the 5x5 grid -- something different was happening with that grid than with the 4x4 grid.  But what?

After another minute or two, a girl raised her hand and told me she had counted white and shaded squares.  (This was the right idea!)

With all the children, we counted 13 shaded squares, but only 12 white squares.

After another minute, I asked the children to tell me what colored squares my path stepped through.  If I started on a shaded square, where would I go next? 

To a white square.

Why not a shaded square? 

Because you can't move diagonally.

So then we realized that if your path started on a shaded square, it would proceed as follows:
Shaded - white - shaded - white -shaded - white - ... through 25 squares (in the 5x5 grid case).

If you started on a shaded square, where would it end?

After a minute or so, they figured out that it would end on a shaded square if there were 25 squares.

Then we counted.  That meant it would run through 13 shaded squares, and 12 white ones.  Hey!  That's how many we had!

Then I talked about a path that started on a white square.  Where would it end?

We stepped through the path, and it looked like this:
White - shaded - white - shaded - ... - shaded - White!

It ended on a white square.

"But that's impossible!" shouted one little boy.  "You would have to have 13 white squares!"


We all counted together.  A path that started on a white square would have to go through 13 white squares and 12 black.  But we didn't have 13 white squares, we only had 12 white squares!  That meant that the puzzle I gave the students was impossible!

It was time for cookies then.  But before I let them get a cookie, I told them conspiratorially that they ought to take a copy of the puzzle home and give it to their parents to try. 

They thought that was a hilarious idea, and every one of them came up to get an extra copy of the puzzle.

And to get cookies.


I think this activity worked very well for kids this age.  Because there were different puzzles of different difficulty level, it engaged all the students regardless of age.  Although not all the children seemed to understand all the explanations (especially cycles of even length), they seemed to be having fun and learning something.  I would do this activity again for this age group.   

Saturday, November 2, 2013

Tower of Hanoi

I've been traveling for a few weeks, missing the Saturday math circles.  I'll try to get lesson plans from the other instructors to post here.  Meanwhile, here was this week's activity.

One idea from the following two games is to try to simplify a problem by first examining a simpler problem.

I lined up four chairs in a row, all facing in one direction, and asked for four children to volunteer to help me out.  These children would be standing up and sitting down, following the rules below.

Game rules:
1. The person in front could stand and sit anytime.
2.  Everyone else could only stand or sit when the person right in front of them was standing, but everyone else in front of them was sitting down.

We ran through a few examples to help everyone understand the rules.

Example 1.  Everyone is sitting down.  Who can move?

Answer:  The person in front can stand up, but no one else can move.

Ok.  Now the person in front is standing.  Who can move?

Answer:  The 2nd person can stand up, or the person in front can sit down.  But that's all.

Example 2.  Suppose the first two people are standing up.  Who can move?

Answer:  Be careful!  Some of the children thought that the third person could stand up now, but they can't.  Although the person right in front of the 3rd person is standing, because the 1st person is also standing, the 3rd person can't stand now.

So the only options are the 2nd person can sit down, or the 1st person can sit down.

If the 1st person sits down, then the 3rd person can stand.

After we went through these examples, I had the children try to help me to get the last person, and only the last person, standing.

Goal:  Make the last person be the only one standing.  

A couple of the students caught onto the rules quickly and directed the sitting and standing of the others.  As you try this on your own, you'll notice that the person in front has to stand up and sit down a lot.

We successfully got the 4th person, and only the 4th person, standing with the four original volunteers.  Meanwhile, a lot of new students had come to the class.  I explained the rules again, went over the above examples again, and asked the students to break into groups of four and try it on their own for a few minutes.

(Note:  This was one of those times when I was very glad to have extra parents around.  The parents helped organize the children into three groups of four, and helped to get them started thinking about the problem -- who should be standing and who should be sitting?)

After several minutes, all three groups had been able to get the last person, and only the last person, standing.  I asked them to count how many steps it took them to get that person standing.  One of the groups had already counted, the others hadn't.  For the group that had done the counting, I asked them to figure out how many steps it would take to get the last person standing if there were five people in the row instead of just four.

Everyone worked for a while.  I handed out pencil and paper to those who wanted to use it to help them count.  After a few minutes, when all the groups seemed to have gotten mixed up somewhere, I stopped them.

"How many people are finding this hard?" I asked, and roughly 3/4 of the people in the room raised their hands -- including several parents. 

This seemed like a good time to review our rules of Math Circles, so I had the students help me remember them.  Here is the order in which the students gave the rules:

Rules of Math Circles
1.  Make mistakes.
2.  Have fun
3.  Help others have fun
4.  Ask questions.

I pointed out that a lot of us had already made mistakes, so we were doing the right thing.

Back to the problem:  Now that everyone had a good idea about what the standing and sitting game involved, I asked them to think about a simpler problem.

What if there was only one person in the row?  Was this an easier problem?


How many steps did it take to get one person in the row standing?


And here it is:

What if there are two people in the row?  How many steps did it take to get two people standing?

This is a slightly harder problem, but a few of the children figured it out quickly.  We went over the answer together on the board.

3 steps to get only the last person standing when there are two people.

What about when there are three people?  What about four?  I gave everyone a sheet of paper, and had them try to figure out how many steps this would take.

While they worked, I walked around the room talking to the children and asking them to explain what they were getting.  Again it was really helpful to have parents around.  A couple of the parents were asking the children questions and helping them to figure out the answers.

After about five minutes, I called everyone together and asked for answers.  The children had figured out the following.

3 people in the row:  7 steps
4 people:  15 steps
5 people:  31 steps  (not all the groups had figured out this one)

A couple of the groups were trying to figure out a pattern.  I gave them a hint.

When there are five people in the row, what does the row have to look like before the last person can stand up?

Answer:  The 4th person, and only the 4th person, must be standing in order for the last person to be able to stand.

I put a picture like the one above on the board.  Then I put my hand over the 5th person in the row.

Ok.  Notice that in order to get the 5th person standing, you first need to get the 4th person, and only the 4th person standing.  But we just figured out how many steps it takes to get the 4th person, and only the 4th person standing.  Right?  How many steps?

Several of the children realized at this point that this was the solution to the previous problem.  It took 15 steps.  With that hint, I asked them to see if they could figure out a pattern, and if so, figure out how many steps when there were 6 people, 7 people, and 10 people.

Again I gave them about five minutes.

One little boy figured it out quickly on his own after I repeated my hint again.

"How many steps does it take to get the 4th person, and only the 4th person, standing?" I asked.

"15 steps," he said.

"And then when the 5th person stands up, how many steps is that?"

"One more," he said.

"And then are we done?" I asked.

"No," he said.  "You need to get everyone else sitting down."  And then he thought for a second.  "And that will take 15 more steps!" he concluded.

Meanwhile, a couple of groups had figured out a pattern:  double the last number and add one.  About 2/3 of the students were still following, having fun doubling numbers and adding one.  The others were lost or distracted.  I tried to help those who weren't following for a couple of minutes, but by now it was time to move onto a new game.  I asked those who had finished to give me the numbers of steps.  Here they are.

6 people:  63 steps
7 people:  127 steps
10 people:  1023 steps


You can read about the Tower of Hanoi on Wikipedia, for example.

Basically, you have a stack of disks, each of a different size, and three pegs.  You move the disks between the pegs, according to the following rules. 

Tower of Hanoi Rules:
1. You can move only one disk at a time.
2.  A disk can never be moved on top of a smaller disk. 

I gave each child four paper disks in four sizes, and had them make three X's on their paper for the pegs.  We went through an example on some legal moves on the board.

If all the disks are stacked up, what is the first move that we make?

We move the small one to one of the other X's.

Now we want to move the next smallest circle (red in the figure).  But it can't go on top of the smallest circle, so it has to go to the other X.

"Oh, this is easy!"  shouted one boy.

Goal:  Move all the disks from one X to another, following the rules.

I let them work on their own for a few minutes.  All the children were interested again and playing.  After walking around a bit, I noticed that a couple of students were moving more than one piece at a time, so I reminded them that they had to move only one disk at a time.

"Oh, that's hard!" said the same boy who had declared it was easy a moment ago.

Nevertheless, after a few minutes he and the girl next to him had finished.  After a few other children had finished, I asked everyone to count how many steps it took to move the whole stack of disks. 

A couple of students raised their hands to show me how they had solved the problem.  I watched one boy show me how to move the stack in 16 steps.  Another could do it in 17 steps.  One little girl was excited to show me how to move the stack in 14 steps, but it turned out that her solution really used 15 steps.

At this point, we were nearly out of time, so I called everyone together.

"What if we only had one circle?" I asked.  "How many steps to move that circle to another peg?"


I wrote "One circle" on the board next to "One person" from the previous standing/sitting game.

One circle:  1 step.

"What if we had two circles?"

After a couple of seconds, a few children shouted out:

Three steps!

Two circles:  3 steps.

"What if there are three circles?"

There was silence for longer at this point, while some of the children tried to quickly figure it out.  One little boy in the corner was prompted by his mother to raise his hand, so I called on him.

7 steps.

Now the board looked something like this:

Standing/ Sitting game     Towers of Hanoi
1 person 1 step 1 circle 1 step
2 people 3 steps 2 circles 3 steps
3 people 7 steps 3 circles 7 steps
4 people 15 steps
5 people 31 steps

We were out of time, but I told the children to think about the patterns and see if they could figure out what happened at home, and why. 

As we distributed cookies, a couple children came to me and told me excitedly how many steps they had needed to move circles.  Over all, they seemed to have had fun and to have learned something. 

Monday, October 7, 2013

Tic-tac-toe and other games

Dr Pace Nielsen, a professor at BYU, led our most recent math circle.  His focus was games on grids, like tic-tac-toe, but with interesting modifications.

Preparation:  Each student needs a pencil, and two or three sheets of graph paper.

Introduction:  Dr Nielsen reminded everyone how you play tic-tac-toe on a 3x3 grid.  The first player places an X in one of the squares.  The second player places an O.  When one player gets three of their symbol in a row, either horizontally, vertically, or diagonally, that player wins.  If the grid is filled and no one wins, we say "the Cat wins". 

A child was invited to the board to play a couple of games.  The children soon saw that if each player used their best strategy, the Cat would always win in the 3x3 game. 

Activity 1:  Each child received graph paper and a pencil.  With another child, they were asked to play tic-tac-toe, only with modified rules.  Instead of playing on a 3x3 grid, they would play on a 4x4 grid.  The same rules applied:  X starts, O goes second.  The first child to get three in a row, either vertically, horizontally, or diagonally, would win.  Note that even though the grid size is 4x4, the players still only needed three in a row to win.

Example 4x4 Tic-Tac-Toe games played by two children.  Note O won a couple of rounds, until X figured out a strategy....

The children played several rounds for several minutes, switching who played X and who played O.  Dr Nielsen and the other classroom helpers walked around a bit, observing the children's games, sometimes giving hints.  After they had had the chance to play for a while, Dr Nielsen called everyone back to ask about strategy.

"What happens in the 4x4 tic-tac-toe game?  Does the Cat win?"

By now, most of the groups had discovered that someone, either X or O, seemed to always win, although not all the groups had figured out a strategy.  However, one enthusiastic little girl raised her hand high.

"X always wins if they are smart!"

Dr Nielsen then asked the children to help him walk through a strategy that would guarantee that X would win.  First move:  X should start in the middle.  Then the children noticed that no matter where O went, X could go again in the middle, adjacent to their first X, to have two in a row with empty squares on either side.  Then O could not block X from winning the next turn.

Winning strategy for X in 4x4 Tic-Tac-Toe. 

Activity 2:  Now that the children understood 4x4 tic-tac-toe, the rules were mixed up again.  Everything was the same, except they played on a 3x4 grid.  Who would win this time?

After a few minutes of playing, switching between X and O, again a couple of groups had found a strategy where X would always win.  I'm not going to give the winning strategy away this time -- see if you can figure it out.  Hint:  X should start in the middle.

A few 3x4 Tic-Tac-Toe games.

Activity 3:  Infinite tic-tac-toe.  This time, the children were allowed to use a grid as large as they wanted, but they needed to get four in a row.  With a new sheet of graph paper, students played several rounds.

A few minutes into the activity, one girl started jumping up and down.

"We figured it out!  X always wins!"

"OK," said Dr Nielsen.  "Let's play."  He sat down with the girl and they played together.  The first round, he won, but she made a mistake, and wanted to play again.  The second round, he also won.

"But guess what?" he said confidentially.  "You are right!  X always wins."

"If they are smart," said the girl, meaning X wins if they know what they are doing.

Dr Nielsen then gathered the group together then and asked them who wins.  Those who had overheard the earlier exchange knew that X would win, and he told them that was correct.  However, the game was much harder than the others they had been playing. 

"So if you're tired of regular tic-tac-toe, and you want a challenge, play infinite tic-tac-toe instead."

Activity 4:  Angels and Devils.  At this point, everyone got a new sheet of graph paper and we started a new game, called Angels and Devils.  In this game, X and O take turns, just as in tic-tac-toe.  However, rules for movement and objectives are different.  X must start in the middle of the sheet of paper, and on their next move, they can only go into an adjacent square.  O, on the other hand, can go anywhere.  Their goal is to keep X from getting to the edge of the paper.  If X makes it to the edge of the paper, then X wins.  If they are blocked, O wins.

Again the children played for a few minutes.  One group was convinced that X would always win.  Dr Nielsen came to check out their solutions and noticed that they were dividing their graph paper into small blocks.  "Try playing on a much larger grid," he suggested. 

In fact, mathematicians have been able to show that X will actually always lose when the rules are as given above.  (See, for example, this wikipedia page.)  While none of the younger children were able to come to this conclusion in the amount of time available, they had fun drawing X's and O's for a while.  When some of them started getting restless, we switched activities again.

Activity 5:  Tapping fingers.   Before starting the new game, Dr Nielsen had everyone switch tables and pick new partners.  He then called up a student to help him demonstrate the rules of the new game, which was a little different.  This game doesn't require pencil or papers, but just a partner and hands!

Both he and the student held out two hands, with one finger extended on each hand.  The first player tapped one of the hands of the other player with one of their hands.  The player who was tapped needed to add a number of fingers equal to the number that was on the hand of the other player who tapped them.

For example, on the first round, when the first player had 1 finger on the right hand, 1 on the left, and the second player had 1 on the right and 1 on the left, no matter which hand the first player tapped, the second player would end up with 1 finger on one hand, and 2 on the other.

Now the second player takes a turn.  They can either tap the first player's hand with 1 finger or with 3.  If they tap with 1, the first player will add 1 finger to the hand they tap.  If they tap with 3, the first player will add 3 fingers to the hand they tap.

For example, now the second player has 2 fingers on his left hand, 1 on his right, and the first player has 1 finger on each hand.  The second player tapped the first player's left hand with his 2-finger hand.  Then the first player had 3 fingers on the left, and 1 finger on the right.

If a player is tapped by another so that the fingers on the hand add up to 5, that hand is dead, and must go away.

However, if the fingers add up to more than five, for example if a hand with four fingers taps a hand with three fingers, then the 3-finger hand gets all four added as follows:  first add 2 -- now there are five fingers and the hand closes.  Then add the remaining 2.  These stick.  So a hand with 3 fingers tapped by a hand with 4 fingers gives a hand with 2 fingers.  (Is that confusing?  With some help the first time it came up, even the kindergarteners were able to figure it out.)

One last rule.  If one of your hands is dead, but there are an even number of fingers on your other hand (either 2 or 4), you can use your turn to tap your dead hand with your other hand, and split the fingers between the two hands.  So if you tap your own dead hand with a hand with 2 fingers, each hand ends up with 1 finger.  Similarly, if you tap your own dead hand with a hand with 4 fingers, each hand ends up with 2 fingers.

The first player to have two dead hands loses.

Once the rules were explained, and demonstrated, the children paired up and played the game together.  I hadn't been playing the tic-tac-toe games, but this time I got a partner -- a kindergartener named Eleanor.  Eleanor wasn't in a super competitive mood, and neither was I, so we took turns tapping hands back and forth without really killing off any hands.  While we weren't coming up with a winning strategy, Eleanor was still having fun practicing simple addition (and modular arithmetic!), and she was pretty good at it.

Eleanor's mom, who is a friend of mine and apparently somewhat more competitive than her daughter, began giving Eleanor suggestions for how to defeat me.  Eventually, with Eleanor's laughing support, the mom took over Eleanor's hands and began directing operations.  We were having lots of fun, but we still hadn't figured out a winning strategy before time was called...  Cookies!

Because we were having too much fun, I missed Dr Nielsen's concluding remarks.  You will have to figure out your own winning strategy for this game.  

And next time, Eleanor and I are going to gang up on Eleanor's mom. 

Monday, September 23, 2013

Fractal triangles

Dr Emily Evans, a math professor at BYU, led the activities for September 21. 


Dr Evans' activity was based on one on the website

She printed several copies of triangles from that website, in two versions.  The easy version, for the younger children, had lines filled into the triangles.  The more difficult version, for the older children, had just the outline of the triangle.  Both are available on that website, along with lots of great pictures that can supplement the lesson and illustrate what's going on. 

Dr Evans also brought crayons, markers, and pencils for coloring triangles, and scissors for cutting them out.

In addition, she brought examples of fractals:  two ferns she picked up at  flower shop, and a book with pictures of mountains, snowflakes, etc.  


Dr Evans welcomed the children to Math Circles.  She requested that the younger children sit on one side of the room, and the older ones on the other side, so they were pretty well sorted by ability at the beginning.

She asked students who could remember the Math Circles rules, and we went over them one more time:

1.  Make mistakes 
2.  Ask questions
3.  Have fun
4.  Help others have fun.

At this point, Dr Evans explained what a fractal is.  A fractal is an object that is "self-similar":  it contains a pattern that repeats again and again.  She brought in several examples.

First example:  A fern.    Note that a fern has a stem with many stems growing out of it.  Each of those stems also has many stems growing out of it.  At the next level down, the veins in the leaves radiate out of the stem just as in the two previous levels, and so on.  She passed around the two ferns she had brought. 
A student showing the ferns, with Dr Evans in the background.  Note also the fractal pyramid!
She then asked if anyone knew of other examples of fractals.  With some prompting, a girl suggested a mountain, and we talked about how the jagged shape of the mountain is repeated in the jagged shapes of rocks on the mountain, and the jagged bits of rocks, and so on.  We also talked about the shore of a lake, and the fractal pyramid Dr Evans brought along.


After the discussion, Dr Evans handed out the papers she had copied with triangles on them.  Those who had a blank triangle would be drawing a fractal on their paper.  Those who already had a fractal inside their triangle would be coloring.

The picture above shows one student's first two levels in creating the fractal.  First, an upside down triangle is drawn by connecting midpoints of the original large triangle.  This leaves three upright triangles.  Find the midpoints of each of these, and connect them.  This leaves nine upright triangles.  Find midpoints of these and connect them.  Students can keep drawing triangles as long as they would like.

... And many of the children did draw triangles for a long time....

After the triangles were finished, the students were instructed to cut them out. 

About 30 minutes into the lesson, Dr Evans took a group of those who were finished out into the hallway where there was a wide space to build a fractal out of the triangles that the students had colored and decorated.  She asked some of the older ones to help her form a new, larger fractal.  (Those who were still coloring stayed in the classroom -- remember how we recommended having extra adults for kids this age?  This is one place where that really helped.)

Assembling the fractal in the hallway is tricky.  Many students just want to stack triangles on top of each other. You can point out to them that they are making a mistake!  Which is a good thing -- they're following the rules.  But then they need to figure out how to fix the mistake. 

Make sure they are building the triangles in a repeating pattern.  The upside down triangles that were drawn in the steps above will be carpet when you spread the triangles on the floor.  The smaller white triangles (with fractals drawn on them), will fill in the fractal. 

In the above photo, the children have placed a few more triangle.  In our class, it was really just two or three of the older children who got excited about placing the triangles on the floor.  A lot of the others were very happy to continue coloring for a long time.

When the fractal was all done, the children lined up in a row on the bench behind their creation, and parents took pictures. 

By then, 50 minutes had passed, so we reconvened in the classroom for cookies.