## 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

## Christopher Long’s fun generalization of an Expii problem

Twitter is really great place to see fun math. Before showing the fun generalization, though, just to avoid spoilers I want to show the original problem. Here’s the direct link:

https://www.expii.com/solve/69/5

and here’s the problem itself:

So, I’ll pause here to not ruin the problem for anyone who wants to work on it.

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Ok . . . here’s the really cool set of tweets I saw from Christopher Long this morning:

which continued as follows:

To see how delightful Long’s general solution is, maybe walking through my chicken scratch solution which happened to be still sitting on my desk will be helpful:

Here’s a sketch of my approach.

In order to maximize your chance of winning a bet like the one in the problem (one that you expect to lose) you should bet as much as you can at each step (subject to a maximum of the amount you need to win) at each stage.

So,

(i) at step 1 the probability of getting the tree to grow to 40 feet is 1/5 and probability of losing is 4/5.

(ii) Assuming you win, you now have a 1/5 probability of getting the tree to grow to 80 feet and a 4/5 probability of losing.

(iii) Assuming you win, you now have a 1/5 probability of getting the tree to grow to 100 feet and a 4/5 probability of having it shrink to 60 feet.

(iv) If you win on stage (iii) you win (1 out of 125 times). If you lose, you now have a 1/5 probability of having the tree now grow to 100 feet and a 4/5 probability of having the tree shrink to 20 feet.

So, after the 4th branch in my picture you’ve either won (probability 1/125 + 4/625), returned to 20 feet (probability 16/625) or lost (the only other case).

Thus, your probability of winning the game from the start, $x$, satisfies the equation:

$x = 1/125 + 4/625 + (16/625)*x$

We can solve this pretty easily to see that $x = 3/203$.

The really fun – and honestly, amazing – thing about Long’s solution is that he notices that the pattern in the branches of the binary tree corresponds exactly to the pattern in the digits of the binary expansion of 1/5. For clarity, the 1/5 here comes from the growth multiple – 20 feet growing to 100 feet – and not from the probability which, by coincidence, also has a 1/5 in it.

Anyway, Long’s solution also allows you to immediately see how to solve any problem like the Expii one, and, for extra fun, problems where the growth multiple is irrational:

The answer is in Long’s timeline, but it is a good challenge to see if you can work out the answer just from the tweets I’ve included here. Since he skips a bit of algebra in his tweets, working through his tweets is also an important way to make sure that you really understand his work.

I think the sequence of tweets from Long are a great thing to show kids who are learning math – especially kids learning probability and stats. Those tweets really show how a mathematician thinks about a problem.

## 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:

## Sharing problem #3 from the European Girls’ Math Olympiad with kids

Yesterday I saw the great news that team USA won the European Girls’ Math Olympiad:

Flipping through the problems last night, problem #3 really caught my eye as one that math students might really enjoy because the solution is really cool. Here’s the problem:

This afternoon I thought it would be fun to talk through the problem with the boys. I have no expectation that they would be able to solve this problem – obviously! – but I really did think that a sketch of the solution would be really interesting to them.

I started by talking through the problem to make sure that they would understand it:

Once the boys understood the problem we dove into trying to solve it – where do you even begin – both boys said in the last video that the problem seemed impossible! Starting with some simple configurations with 2, 3, and 4 lines helped us see that the answer to the problem might be “no”.

To wrap up I showed the boys how you solve this problem via a coloring argument. The critical idea is that you can color the regions that are formed by the lines, with no two regions sharing a side having the same color – with just two colors. Once you have the coloring, there’s a fun little “aha” moment when you watch the path the snail takes . . . .

So, a seemingly impossible problem has a really pretty and really instructive solution. I think the coloring idea is something that middle school and high school kids who are interested in math will really enjoy seeing.

## A project inspired by an AMC 12 octagon problem

The problem pictured below from the 2003 AMC 12 gave my son some trouble:

We talked through it together a few days ago, but I thought it would be fun to try to do an octagon-inspired math project today.

We started with the problem and then talked a bit about a 3d print we found on Thingiverse:

Next we took a look at a version of the 3d printed shape that we made from our Zometool set. You can’t make a regular octagon with a Zometool set, and the fact that our shape didn’t have a regular octagon led to a good discussion:

For the last part of the project we tried to find the volume of our truncated cube.

## An AMC12 algebra problem that gave my son trouble

The problem below gave my son some trouble this morning:

When he got home from school we talked about it in more detail and it seemed to make more sense for him than it did this morning. The problem is a nice introductory algebra / quadratic problem:

Next I showed him a similar solution, but where “x” represented a different number:

Finally – just for a completely different way of looking at the problem – I wanted to show him a way that we could use the choices to help us find the solution. This is sort of cheating, but he was very confused by the problem this morning and I wanted to show him a way to get a little un-stuck when you are stuck.

Also, we got interrupted by the guy servicing our furnace – so sorry the video jumps in the middle 🙂

## A challenge relating to a few problems giving my son trouble

I’ve seen some interesting ideas from Tracy Johnston Zager over the last week about the relationship between learning math and intuition. For example:

Although I’ve been traveling a bit for work this week the relationship between learning math and intuition has stayed in my head. Sometimes my thoughts have drifted to and old blog post about a problem from the European Girls’ Math Olympaid:

A Challenge / Plea to math folks

That post, in turn, was inspired by an old post by Tim Gowers where he “live blogged” his work while he solved a problem from the International Mathematics Olympiad.

It can be really hard for anyone to know what math intuition looks like because everyone sees polished solutions way more often than they see the actual process of doing math.

That’s part of the reason I make the “what a kid learning math can look like” posts – so everyone can see that the path kids (or anyone!) actually takes to the solution of a problem is hardly ever a straight line:

Our “what a kid learning math can look like” series

The other thing on my mind this week has been some old AMC 10 problems that have really given my older son some trouble. These are pretty challenging problems and require quite a bit of mathematical intuition to solve.

So, I’d like to make the same challenge with these problems that I made with the problem from the European Girls’ Math Olympiad – “live blog” yourself solving one of these problems. Post the though process rather than a perfect solution. Let people see *where* your mathematical intuition came into play.

## Struggling through a challenging AMC 10 problem

This problem gave my son some difficulty yesterday – it is problem #19 from the 2011 AMC 10a

Last night we talked through the problem. The talk took a while, but I was happy to have him slowly see the path to the solution. Here’s his initial look at the problem:

Next we looked at the equation $y^2 - x^2 = 141$. Solving this equation in integers is a nice lesson in factoring. Unfortunately by working a bit too quickly he goes down a wrong fork for a little bit.

In the last video we found that the original population of the town might have been 484, and it might have changed to 634 and then once more to 784. We had to check if 784 was a perfect square.

Finally, we needed to compute the approximate value (as a percent) of 300 / 484. The final step in this problem is a great exercise in estimating.

So, a really challenging problem, but also a great problem to learn from. We went through it one more time this morning just to make sure that some of the lessons had sunk in.

## Why I love watching my kids learn math

Had a great night with the boys tonight. My older son was working on some old AMC 10 problems and we talked through one that stumped him for his movie:

It was #15 from the 2013 AMC 10a:

Next I spent some time with my younger son. He’s been studying the basics of lines using Art of Problem Solving’s Introduction to Algebra book plus a little bit of Khan Academy (when I’ve been traveling for work). I asked him what he’d learned so far and loved his response. It was a great reminder of the joy of learning new math ideas for the first time:

## Struggling with a challenging AMC 10 geometry problem

This problem from the 2016 AMC 10 a gave my son some trouble yesterday:

We talked through it last night and it was interesting to see where his intuition was off:

After finding a solution in the last video that he knew was wrong (because it wasn’t one of the choices on the test) we looked back to see if there was alternate approach to the problem. This approach led us to find the actual solution and also the mistake in the first approach:

It is always fascinating to the though process on a challenging problem. Sometimes the thought process is so close to the right approach that the mistake is really tricky to spot.