Tag counting

Playing with Jim Propp’s essay on Arthur Engel

Jim Propp’s August 2017 blog post is absolutely terrific:

Prof. Engel’s Marvelously Improbably Machines

Even though we are visiting my parents in Omaha, I couldn’t resist having the boys watch the video in the essay and then play with the challenge problem.

Here’s what they thought after watching the video – the nice thing is that they were able to understand the problem Propp was discussing [also, I shot these videos with my phone, so they probably don’t have quote the quality or stability of our usual math videos]:

Next I had them play the game that Propp explained in his video. The idea here was to make sure that they understood how the [amazing!] solution to the problem shown in the video:

Next we tried the challenge problem from the essay. I almost didn’t do this part of the project, but I’m glad I did. It turned out that there were a few ideas in the Propp’s video that the boys thought they understood but there was a bit more explanation required. Once they got past those small stumbling blocks, they were able to solve the problem.

I’m really excited to dive a little deeper into the method of solving probability problems that Propp explains in his essay. What makes me the most excited is that the method came from someone thinking about how to explain probability to kids.

The last video shows that understanding Engel’s method does take a little time. Once kids get the general idea, though, I think they’ll find that applying to a wide variety of problems is pretty easy. It is amazing how such a simple method can made fairly complex probability problems accessible to kids.

The “Dungeons and Dragons” problem

My kids have suddenly been drawn into D&D. They are having a ton of fun with it and I thought that there was at least one fun little math problem we could talk through that related to the game.

At the start of the game you create a character with various properties. To determine the value of some (maybe all) of those properties you roll four 6-sided dice and add up the three highest numbers.

The question we looked at today was what is the expected value of that sum?

First we introduced the problem and come up with a few ideas about what the answer might be.

Next we did 10 trials to see what average we’d find:

Now we had a longish talk about how you might solve the problem. The boys jumped to a computer simulation pretty quickly. After talking about how that simulation would work we talked about how to solve a similar problem with two dice.

Finally, we did go the computer to see what the answer would be. Talking about how to write the program was pretty fun.

Nice project – I might revisit this one to talk through the geometry of the solution of the 2 and 3 dice problem and see if the boys can figure out how to generalize to the 4 dice case.

Working with the PCMI books part 2: coloring an octahedron

Last week we got the PCMI books:

Our first project involved a neat problem about understanding the number 0.002002… in different bases:

Playing around with the PCMI books

Today I was looking for another fun problem and found another problem that I thought would make a fun project:

Barbara has an octahedron, and she wants to color its vertices with two different colors. How many different colorings are possible? By “different” we mean that you can’t make one look like the other throu a re-orientation.

I started by introducing the problem and asking the kids what their initial ideas were:

They had a couple of pretty good ideas including some basic ideas about symmetry. Using those ideas we began counting the different colorings:

We counted the cases in which 3 vertices were black and 3 vertices were red. This case proved to be tricky, but going through it slowly got us to the correct answer.

Finally, as a fun little extension, I asked them to find the number of ways to color the faces of a cube with two colors. Having solved the octahedron problem already, this one went pretty quickly, and they even noticed the connection between the two problems 🙂

I like this problem. I’m glad that the boys were able to see some of the basic ideas. When you add more colors the counting gets much more difficult and some pretty advanced math comes into play. The number of colorings with “n” colors is:

(n^6 + 3n^4 + 12n^3 + 8n^2) / 24

The different terms correspond to different symmetries of the cube / octahedron. We’ll have to wait a few more years to cover the complete details 🙂

The coupon collection problem with kids

Yesterday my younger son was playing a dice game (explained in the first video) that reminded me a bit of the coupon collection problem. I thought it would be fun to try out that problem with the boys this morning. We were a little low energy, but I think it was still a good project. I’ll have to figure out how to revisit it to make sure the points stuck.

Here’s the introduction, including the game my son was playing:

Next we worked through one case of the problem – rolling dice trying to collect 6 “coupons”. My older son thought it would take 15 rolls and my younger son thought it was take 20.

Now I tried to help the kids dive into the math. We ended up going down a path that was much more complicated than I intended. I’m not sure why I made the choice that I did here, but . . . it happens sometimes 🙂

So, at the end of the last video we were caught in a seemingly complicated infinite series. I tried to explain why the expression we had on the board had to be equal to one. Then I tried to explain why the expected number of rolls had to be greater than one. The explanation here is a disaster, though.

Now that things had gone totally off the rails, I tried to pull it back. Luckily things did go better, and it was easier for the boys to see the expected number of rolls when there were fewer open slots.

Finally I wanted to show the kids how the ideas we talked about here would apply to a more difficult problem – say 100 coupons. We got off on the wrong foot here, but we eventually saw how the ideas we’d talked about previously applied.

Despite the low energy and going doing a path that was a bit too complicated, I think this is a fun problem for kids to study. It looks very difficult initially, but through a bit of calculation (and maybe a bit of hand waving) we can break it down into some smaller problems that we are able to solve. Putting the solutions of those smaller problems together, we can show that the solution to the original coupon collection problem isn’t too hard to understand.

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A good (though tricky) introductory counting problem

My older son is re-working his way through Art of Problem Solving’s Introduction to Counting and Probability. He came across a problem in the review section for chapter 5 that gave him some trouble. I decided to talk through part of it tonight and included my younger son.

My younger son hasn’t been studying any counting lately, so I was expecting the problem to be pretty challenging for him. His work through the first part of the problem is, I think, a nice example of a kid working through a challenging math problem.

The problem is this: How many different ways are there to put 4 distinguishable balls into 3 distinguishable boxes?

The next problem is what was giving my son some trouble: How many different ways are there to put 4 distinguishable balls into 3 indistinguishable boxes?

My younger son struggles with the problem for a bit on this one and then my older son offers his thoughts. What gives my older son a little trouble is the case in which you put 2 balls into one box and 2 into another.

So, after struggling with 2-2-0 case in the last video, we talk about it in a little more depth here. The tricky part is seeing that two cases that don’t look the same are actually the same. It was harder for the boys to see the over counting than I thought it would be. But, we made it!

So, the indistinguishable counting part was pretty confusing to the boys. I think we need to do a few more problems like this to let this particular counting concept sink in.

James Tanton’s counting problem part 2

Yesterday we looked at a really neat problem James Tanton posted last week:

That project is here:

Working through a challenging counting problem from James Tanton

Our first look at the question involved some dice rolling and a computer simulation. Today we are going to look at an exact solution to the problem. That solution involves studying all of the different things that can happen when you roll 5 dice. It turns out that there are 7 different patterns that can happen, and these patterns related to the ways you can write 5 as the sum of positive integers.

(1) 5 different numbers, which I’ll represent as 1 + 1 + 1 + 1 + 1

(2) 3 different and 2 the same -> 1 + 1 + 1 + 2

(3) 2 different and 3 the same -> 1 + 1 + 3

(4) 1 different and 4 the same -> 1 + 4

(5) 1 different and 2 pairs -> 1 + 2 + 2

(6) 1 pair and 1 triple -> 2 + 3

(7) All numbers the same -> 5

For today’s project we’ll count the number of ways that each of these 7 patterns can occur. We know that the total number of arrangements is 7,776, so that’s going to help us make sure we have counted correctly.

Here’s the introduction to the problem and to the approach we are going to take today:

Now we began to count some of the arrangements. In this video we count the number of dice rolls in (1), (4), and (7) above:

Now we moved on to some of the more challenging arrangements. Here we looked at (6) and (2) above:

Now we looked at case (5). This case proved challenging because dealing with the 2 pairs caused a little confusion between over counting and under counting. But, after looking at the cases carefully we did manage to get to the answer.

At this point we had only one case left -> (3) from above. But, the counting practice that we’d had up to this point helped this case go pretty quickly.

Finally, we added up our numbers and checked that we’d found all 7,776 cases. We did!

The one thing left to do was to count the different numbers that we saw in each case and find the average. I’d done that ahead of time just to save a bit of time in the movie. Our final answer was (27,906) / (7,776) or about 3.588. The exact answer was (happily!) very close to the two estimates that we had found in our simulations yesterday.

I love Tanton’s problem. It is a great estimation problem as well as a great counting problem. We might do one more project tonight on yet a different way to solve the problem using Markov chains:

Looks like a fun idea – I’ll be thinking about how to talk through this approach with the kids during the day today.

Working through a challenging counting problem from James Tanton

I saw a neat problem from James Tanton during the week:

A follow up post from Amy Hogan made me realize that I should use the problem for a project:

So, I decided to make looking at this problem our weekend project. Today I wanted to try a few simulations. Tomorrow we’ll count the cases. That case work is pretty challenging, but I think we’ll be able to get through it.

Here’s the introduction to the problem:

Next I had the boys roll 5 dice 50 times and record the numbers they saw each time. Here are those results as well as our first estimate of the answer to Tanton’s question:

Finally we moved on to do a simulation. Normally I would have used Mathematica, but I’ve got a program related to one of our old Goldbach Conjecture videos running right now. This problem was easy enough to deal with on Excel, though, so I looked at a simulation there.

Here’s what they saw in the numbers:

Definitely a nice start to this project. Counting the different cases tomorrow will be a bit more difficult, but we’ll also get to talk about some pretty interesting mathematical ideas like partitions and choosing numbers. I’m excited to see if we are able to get all the way to the end.

A challenging counting problem for kids learning algebra

My son is in a weekend enrichment math program and that program has been great for him. It comes to an end this week. The last problem on this week’s homework assignment gave him some trouble, so I thought it would be fun to see if we could work through it together.

I was a little worried because I’d not seen the problem until just before the project, but luckily things went ok.

Here’s the problem:

a, b, c, and d are positive integers less than 10. How many solutions are there to the equation a + bcd = ab + cd?

[post publication note: Originally the text presented the problem incorrectly. It is correct in the videos. Karen Carlson pointed out the typo to me. Sorry about that.]

Here’s how we got started – my son had found several cases, but not quite all of them:

After the introduction to the problem and my son’s work so far, we moved on to try to find more solutions. The main idea I gave my son involved writing the equation in a slightly different form:

Now that we had a plan, we moved on to counting the rest of the cases that we found in the last video:

Finally, we went to Mathematica to write a little program to count the solutions for us. This part of our project turned out to be more interesting than I was expecting. It was interesting to compare the brute force solution of the computer to the case by case counting technique that we’d just gone through.

So, a fun problem that definitely made my son think this week. It is

Studying shuffling and Shannon entropy part 2

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We did a fun project about Shannon Entropy and Shuffling yesterday:

Chard Shuffling and Shannon Entropy

That project was based largely on an old Stackexchange post (well, comment) here:

See the first comment on this Stackexchange post

Today I wanted to extend that project a little bit and thought it would be fun to look at a different kind of shuffle to see if there was a difference in entropy.

Here’s the shuffle for today as well as what the boys think will happen with this shuffle:

Next we took some time off camera to enter the card numbers in our spreadsheet. Here’s what we found for the new entropy after one of these new shuffles:

Finally, we took even more time off camera to do 4 more shuffles and write down the order of all the cards. After that we did 5 successive shuffles and wrote down the numbers after the 10th shuffle.

The kids didn’t think the cards were as mixed as they were in yesterday’s project, and here’s what the entropy calculation said:

I really enjoyed these two projects. It was especially fun to see how kids could get their arms around the idea of entropy even though the math itself is pretty advanced.

A neat expected value problem from Expii

[sorry for the quick write up – I got asked to help out with my son’s archery class today, so I just decided to publish this one as it was when I get asked to help . . . ]

I saw a neat expected value problem from Expii yesterday. In case you’ve not see their site, here’s the link to their main site:

Expii’s front page

and here’s a direct link to the problem:

A neat expected value problem from Expii

The problem goes like this:

“You are planting some trees as environmental action for Earth Day. At each of 200 spots around a circle, you place a seed. Each seed will sprout into a small tree with probability 1/2. Sadly, some of these small trees will die. In particular, a small tree dies if it has another small tree as its neighbor, because they will be fighting for sunlight.

What is the expected value of the number of trees that are still alive at the end of the year?”

I thought this would be a great problem to discuss with the boys. We just got back from a vacation in San Diego and my younger son was still on west coast time, though, so I just talked through this one with my older son.

First I introduced the problem and we double checked that he understood it:

Next we discussed some simple cases to see if we could get our arms around the problem:

Now we moved on to the general case. My son understood some of the main ideas about the problem, but made a small mistake at the end that led to a very small expected value.

Finally, we wrapped up by looking at the error at the end of the last video and trying to calculate the expected value slightly more carefully: