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Month October 2014

A 3D geometry proof with few words courtesy of Fawn Nguyen

Fawn Nguyen is so great at sharing interesting math problems.  On twitter, on her blog, in person, and probably some bat signal type thing if you live in California – she’s got such a great nose for interesting math.   By super happy coincidence earlier in the week we were working through oddly similar problems.  I wrote about it here:

When My Evening was Similar to Fawn Nguyen’s Evening

This morning I finally got around to doing a quick version of the problem Fawn shared with my kids.

The problem is this: how do you find the volume of a Rhombic Dodecahedron, or as she put it on Twitter:

Here’s the start, just showing what the shape looks like:

Up next is this fun observation from Fawn:

The final step is figuring out the volume of all of those pyramids – you must must must need a formula, after all, this is math we are talking about, right 🙂

I wish I’d been filming the construction of the pyramids inside of the cube, because my younger son’s face when he realized that the 6 pyramids filled the cube was priceless.

A super fun project that actually can be modified to fill a long time or shorter time.  Really thankful that Fawn shared this one

Fun with abundant numbers

A few weeks ago Tracy Johnston Zager asked about this problem on twitter (and these embedded tweets should be amusing during Halloween week!):

Her post actually played a role in two blog entries so far:

A Neat number theory problem for kids from Tracy Johnston Zager


A neat number theory problem from David Radcliffe

Today’s post was inspired by one of the responses to her question which sort of stuck in the back of my mind:

That response moved from the back of my mind to the front when when my younger son and I got to section 6.4 of Art of Problem Solving’s Introduction to Number Theory book:  Perfect, Abundant, and Deficient Numbers.  Ha – maybe 10 days apart, but a funny coincidence nonetheless.  Time to see if there is indeed fun to be had with these numbers.

We ended up spending two days in this section because there was so much to cover.  As I’ve written a few times recently, finding ways to build up number sense has been on my mind and playing around with factoring (and then adding up the factors) seems like a perfectly fine way to spend time playing around with numbers.   So building number sense was definitely of the goals here.  One other thing that I thought would be fun in this section was talking about why people are interested in these properties of numbers in the first place.

As an aside, I was happy to see this post from Justin Lanier on twitter the other day discussing a time when he and his class worked through one amazingly complicated equation from a theoretical math paper about the Goldbach conjecture:

I think it is fun to try to figure out ways for kids see examples from current math research and I am going to try to put together a special project about prime numbers this weekend after seeing Justin’s post.

Anyway, back to abundant numbers.  To show my son one example where abundant numbers appear in number theory we looked at the Wikipedia page about super abundant number:

Superabundant Numbers

The first equation on that page looks pretty intimidating, but explaining that equation turned out to be a great way to talk about abundant and super abundant numbers with a kid.  Of course, in the background we got lots of good practice with numbers – finding factors, dividing, fractions, and sums.  Yes!!:

Since my son really seemed to enjoy talking about super abundant numbers the next day we went through a similar computation to see if we could find all of the numbers from 1 to 20 that were abundant.  I’m not aware of any really quick way to do this exercise other than checking most of the numbers individually (as my son points out in the video, you don’t have to check the primes).  Although this might be a dull exercise in isolation, he seems to be pretty excited about finding the abundant numbers and he doesn’t appear to find all the arithmetic work here to be dull.

Conclusion:  Twitter was right –  there was indeed some fun to be had with abundant numbers.  Of course, no one would think that a deep dive into the number theory would be appropriate for elementary school kids, but a deep dive isn’t necessary.  As I said, there’s a nice opportunity for building up number sense here and even a lucky opportunity to mention some advanced math.  All in all a neat topic and a fun couple of days!

A day in the life: building and extending number sense

I’m not entirely sure why but I’ve been spending a lot of time recently thinking about different ways to build up number sense.  About a week ago I started a chapter on similar triangles with my older son, and the problems in that chapter have helped me gain a better understanding of the importance of building “algebra sense” (for lack of a better phrase) too.    I’m surprised how many opportunities there are to focus on both of these topics now that I’m actively paying attention to them.   An odd coincidence today made me want to write up the conversations I had with my kids this morning.

But first I want to back up to a coincidence from yesterday.

As I mentioned above I’ve been studying similar triangles with my older son for a week or so.  The bit of math that seems to be giving him the most difficulty isn’t the geometry, it is working with the ratios that arise in the problems about similar trirangles.   Here’s one of the problems we worked through yesterday just to give an example of the ratios that come up in these problems:

I felt that it would be good to review some of the algebra behind these ratio equations before finishing the similar triangle section and found several sets of practice problems on Khan Academy that provided more or less exactly the review I was looking for.  Here’s one set for example:

Khan Academy Ratio problems

Although people have widely differing views about Khan Academy, I think one nice advantage that it has is that the problem sections are great for this type of review.

Interestingly last night Steven Strogatz posted this picture on twitter:

The similarity between the homework I wanted my son to do and the homework assignment in Strogatz’s tweet got me thinking about context.  The motivation to learn more about geometry was enough for my son to understand the purpose of the Khan Academy problems.  Actually, he even asked to do more.   I don’t know the context of the other homework assignment, but do think that without proper context that assignment could seem quite dull.  This coincidence from yesterday reminds me to be careful to be clear about why I’m asking the boys to do the homework I give them.

Now on to today . . . .

This morning my younger son and I were talking about palindromes (section 6.5 in Art of Problem Solving’s Introduction to Number Theory book).  We began with several simple examples – numbers like 11, 454, 34543 – and then he stopped me:

kid:  “I know a long list of palindromes.”
me:  “what is it?”
kid: [ writes the first 4 rows of Pascal’s triangle on the board ]

This example is definitely a fun one for looking at palindromes, but it also turns out to be a great one for building on number sense.  The connection I wanted to focus on was how the rows related to powers of 11, and how that connection seems to break down in the row:  1 5 10 10 5 1.

My first question to him was whether or not this specific row was a palindrome.  He surprised me by saying that although the number you get by putting all of the terms together, namely 15101051, was not a palindrome, you could get a palindrome you looked only at the last digits, so 150051.  Interesting observation.  We’ll have to return to this topic later when we talk about modular arithmetic!

My next question for him was about the powers of 11.  Starting at 11^0, the powers of 11 are 1, 11, 121, 1331, 14641, and 161,051.  Why did we lose the connection to Pascal’s triangle when we computed 11^5?  This led to a wonderful conversation about place value and eventually to showing why we did not actually lose the connection to Pascal’s triangle at all.  Really fun, and I think a neat way to talk through place value while getting in a little arithmetic practice, too.

Later in the morning my older son got tripped up on this problem from the 2006 AMC 8:

2006 AMC 8 problem 24

The problem has a really lucky connection to palindromes since an important observation in solving it is that one number is equal to another number multiplied by 101.  Talking through this problem also led to a good conversation about place value.  Luckily the notes from the conversation about Pascal’s triangle and place value happened to still be up on the board when this second conversation took place.

Seeing some of the earlier work that was on the board my older son said that he thought you could make the row 1 5 10 10 5 1 into a palindrome by working in base 11.  Ha – another unexpected response, but also now a wide open door to talk a little about what I’m calling “algebra sense.”

We quickly reviewed the place value conversation I had with my younger son about how the rows connect to powers of 11, but then looked at what happens in base 11.  Surprise –  powers of 12!!  Don’t think he saw that coming 🙂    Now maybe 5 to 10 minutes of conversation about what the polynomials (x + 1)^n and (x + y)^n look like and we’ve quite unexpectedly done some neat work that helps build up familiarity with algebra and algebraic expressions.

So a fun morning.  As I have the goal of working on number sense in the back of my mind, I’m excited to see all of opportunities that come up to work on it.  Algebra sense, too, but Strogatz’s post from yesterday reminds me to be extra careful about context.  It is fun to take advantage of the lucky times like this morning when that context appears almost by magic!

When my evening was accidentally similar to Fawn Nguyen’s evening

Had a great time with our half group theory, half  Zometool project last night:


Just after I finished writing it up I saw these two posts on Twitter from Fawn Nguyen:


These two posts made me really happy.  First off, we both spent our night figuring out how to put a Zome cube inside a larger geometric structure.  Amazing!!

Even more exciting, though, the specific shape in Fawn’s pictures reminded me of a fun story from this summer.

Every year we head to Cape Cod with a bunch of friends from college – this year we had something like 10 adults and 10 kids in the house.  We knew ahead of time that one of the days was going to be rained out by the remnants of a hurricane and we needed to bring lots of extra indoor activities.  I brought our Zometool set as well as our 3D printer to have some math fun with the kids.

Some of the kids told me that they don’t like math, but since they’ve been on vacation with me lots of times they know that there will be math and fun together.  The 3D printer was a big hit, not surprisingly, and the Zometool set was even more of a hit.  We did a couple of activities out of this Zome Geometry book,

Book Pic

but mostly the kids just played.  One of the 11 year old girls who really does not seem to enjoy school math at all was particularly enthralled by the Zometool set.   She build some wonderful creations on her own including this one:


If you look at the 2nd picture posted by Fawn Nguyen above you’ll see a cube inside a shape known as a “rhombic dodecahedron.”  When you embed the cube inside of this shape you get to see 6 little yellow square pyramids on top of every cube face.  Definitely hard to see that a cube fits inside the rhombic dodecahedron at first, but as I talked about in yesterday’s blog post, the Zometool set is such a great aid because solving these problems is so much better when you can hold the shape in your hand.

Now look at the shape built this summer by the girl who doesn’t like her school math classes.  The important difference here is that the yellow pyramids are inside of the cube instead of on the outside.  Actually she’s divided up the top and bottom of the cube in a clever way so that the inside pyramid equivalent to the outside pyramid in Fawn’s example is built out of several smaller pyramids, but that is a minor detail for purposes of this blog post.  The really neat thing about her shape is that it shows you how to chop up a cube into 6 congruent pyramids, and that observation solves the problem posed in Fawn Nguyen’s first twitter post above!  The volume of the rhombic dodecahedron is exactly the volume of two cubes – the one on the inside plus the one formed by the 6 square pyramids on the outside.  That important second step comes courtesy of an 11 year old kid who doesn’t like math playing around with a Zometool set on a rainy summer morning.  Yay!!

So one shape comes from math teachers’ circle group in California, and the other from a kid just free building, for lack of a better phrase.  One shape from a group dedicated to teaching math, one from a kid who tells me that she doesn’t like math at all.  Funny how fuzzy the boundaries can be sometimes, and amazing how useful the Zometool sets are in helping people see math in a different, and probably more useful, way 🙂

A 3d Geometry project for kids and adults inspired by Kip Thorne

Yesterday I saw an incredible video on a discovery about black holes made by a team led by the physicist Kip Thorne.  The amazing thing about this discovery is that it came about through Thorne’s consulting work in the production of the new movie Interstellar.  You just never know when inspiration is going to strike!  The video I saw yesterday is here, though sorry I can’t get it to embed right:


What really struck me watching the clip above was the statement from Kip Thorne around 2:30 when he describes his reaction to seeing a picture of a properly rendered black hole:  “I’d known it intellectually, but knowing intellectually is completely different than seeing it, than feeling it.”

This thought was in my mind the rest of the day yesterday and I couldn’t stop trying to think up other math-related ideas that would be easier to understand if you were able to hold them in your hand.  At night I was flipping through my old Algebra book from college and found a great example by luck.  This book is one of my all time favorite math books – Artin was a terrific teacher:


In Chapter 7 of the book Artin explains some basic group theory and uses the rotation group of the icosahedron as an example.   One of the theorems shows that the group of rotations of the icosahedron is the same as second group – the alternating group A_5.  The details about A_5 aren’t important for this blog post, but what is important is that a critical geometric idea in the proof is that you can inscribe 5 cubes inside of a dodecahedron.  Here’s the picture Artin gives on page 200 of the book:


I never really completely understood this proof, and I never understood at all how these cubes inside of a dodecahedron related to the rotations of an icosahedron.  Thanks to Kip Thorne’s fun video and our Zometool set, though, I was determined to find out.  The kids were going to come along for the ride, too 🙂

To start off the project I asked the boys to boys to build a large icosahedron out of our Zometool set while I was at work today.  The sides of this icosahedron have length equal to two of the longest blue Zome struts.  Here’s our starting shape:

The dodecahedron is a dual shape of an icosahedron and it turns out there’s an amazing way to add a few new Zome struts to the icosahedron to make a shape that combines the icosahedron with a dodecahedron.  It is really cool to see this shape come together (another Zometool miracle!).  Here is the shape (and this + all of the rest of the videos are published in 1080p HD, so you can watch in hopefully non-blurry full screen to get a better view of the shapes):

With the icosahedron and dodecahedron together, now all we need to do to get to the shape Artin was describing in his book is to add in the cube.  I’m so happy that the Zometool set was able to help with this last step!  With this final shape built, we are holding the picture from the book (plus the original icosahedron) in our hand.  Again, hopefully the cube is visible in the video:

Now to see what the rotations of the icosahedron do to the cube.   Of course this is a fun fact all by itself, and that’s what the boys are seeing.  I’m seeing the critical step in showing that the Icosahedral group is isomorphic to A_5, though, which was the group theory piece that was so hard for me to visualize in college.   With the shape right in front of you it is easy to see how it all works – score one for Kip Thorne!

First we looked at what the rotations of the 20 triangles of the icosahedron do to the cube.  To make what we were rotating a little easier to see on camera we put some lego figures next to each vertex of the triangle we were rotating.  Hope that helps you to see the bottom triangle, and the rotations, a little easier in the video:

Next up, the 5 symmetries that come from the rotations of the icosahedron around a vertex.  As an aside, these 5 rotations sort of help you see the similarities between the icosahedron and the dodecahedron.  Again we lined up lego figures to help you see the rotations a little better in the video:

Finally rotating around and edge.  This is the easiest symmetry of the icosahedron to see since we are just rotating by 180 degrees around the middle of an edge.  It is interesting to see that although the icosahedron itself is unchanged by this rotation, the cube isn’t:

After we finished I asked the kids to help put away all of the Zometool pieces.  My younger son told me that he didn’t want to take apart “this awesome shape” just yet.  Yes!!

I’m glad the boys had as much fun with this project as I did.   I definitely got a better understanding of a piece of group theory that I never really properly understood before.  The boys were able to see (and build!) some amazing 3D shapes and also  play around with rotations and symmetries a little.  So so so much fun!!

MoMath and MegaMenger

Yesterday we visited the Museum of Mathematics in NYC to help out with their MegaMenger build.   The boys had a blast!

Menger Boys
This was our 3rd visit to the Museum and I’m sure there will be many more.  One of the fun attractions is this tricycle with square wheels (sorry for the poor quality of this video):

The MegaMenger project is an incredible project in which people from all over the world are working together to build a giant Menger Sponge out of business cards.  The website for the project is here:


The boys were actually so excited about participating in this project yesterday that we started the day today building a level 2 Menger sponge out of snap cubes.  Although I enjoyed the project, too, I wouldn’t have described my excitement as “build a new level 2 Menger Sponge at 5:30 am the next day excited,” but hey, I’ll take it:

Snap Cube Menger

With that new morning build, there was really  no doubt at all what our Family Math project for the day would be 🙂  We began by simply reviewing our trip to MoMath and some basics about the Menger Sponge.  The specific topics for the day are going to be volume and surface area.  For all but the last movie the questions will revolve around Menger Sponges of ever increasing sizes, like the one being build in the Mega Menger project:

Having touched on the volume of the Menger Sponge in the last movie, we now dive into the volume calculation in more detail.  What I liked here is that each kid had a different way of calculating the volume.  So fun to see the different approaches to counting here!  I showed a third way, too, that has  a sort of surprising twist.  The Level 2 figure we talk about at the end is the shape that we constructed out of the snap cubes that is pictured above:

The surface area calculation is only slightly more tricky.  As with the approach to volume, both boys had different approaches to counting the surface area of the Level 1 Menger Sponge.  It turned out that my younger son’s method was actually the same as mine, so I didn’t add a third counting method here.    Taking through my older son’s direct counting method and my younger son’s method of counting the overlaps was really enjoyable.    We finished by wondering which of these two methods was easier to generalize to the higher level sponges.

Next we attempted to calculate the surface area of the level 2 sponge.  The level 2 sponge is the one that we made out of snap cubes this morning.  Our contribution to the MoMath Mega Menger build amounted to the construction of two of these sponges out of business cards.  The construction from folding business cards took a bit longer than the one from snap cubes, though the business card construction was at 2:00 in the afternoon and was followed by BBQ at Blue Smoke in Manhattan, so maybe I should call it a draw 🙂

The math in this video is the most difficult to follow in this project, but hopefully we work through it slowly enough.  To calculate the surface area of the Level 2 sponge we use the method my younger son suggested in the last video.  We first assume that the surface area is 20 times the surface area of one of the Level 1 sponges (since it takes 20 level 1’s to make a level 2) and then subtract out the surface area that vanishes when two sides touch.  We break down this calculation into two pieces.  The first part is for the middle pieces that touch two other Level 1 sponges, and the second part is for the corner pieces that touch three.  After a 3 minute calculation, we arrive at the surface area of the Level 2 sponge:

Finally – the punch line!  I thought that ending the project with the lengthy calculation above would kill the excitement we had going this morning, so I went in a different direction for the last movie.  Instead of building ever larger sponges, what happens if we start with a sponge of a fixed size and make a Menger Sponge by cutting holes of ever decreasing size in it?    Even thinking about this question may seem strange, but the result is both fun and a little bit perplexing.  Luckily to answer it we can use the numbers we’ve already calculated in the previous videos – we just need to adjust the scale of the sponges.  Adjusting that scale is an interesting lesson all by itself, btw!    After spending a minute or two talking about what tripling the side length of a cube does to a cube’s volume and surface area, we look at the volume and surface area of a the different levels of Menger Sponges with a fixed edge length.  The result is a neat surprise.

So, despite the super early start (!!), we had a really fun morning.  I’m happy that we had a chance to help out the MoMath team with their Mega Menger build.  Hopefully many other kids around the world will get to help out with this project – it is such a great opportunity to hold an amazing math project in your hand.  Exploring the math behind the Menger Sponge seems like a project that lots of kids would love.

Also, if you’ve made it this far and happen to be in the NYC area, head over to MoMath today (Sunday October 26, 2014) to help them finish the build!  And now having finished this morning’s project and written up this blog post by 8:30 am, it is time to take a nap!! ha ha.

A neat number theory problem from David Radcliffe

[sorry if this isn’t edited well, limited editing time due to lots of kid activities today]

Saw this post on twitter earlier in the week:

It made for a great follow up to a previous Family Math which also happened to arise from a problem we saw on twitter:


The first thing that we did this morning was talk through the problem to be clear about a couple of math terms – proper divisors, for example.   We also looked at a few other integers to help make sure that we really did have our arms around the problem.   One neat thing is that the kids were able to see that when we listed the factors of a number, the solution to Dave’s problem required exactly four factors (or exactly two rows of divisors the way were were listing them).

The next step was to try to understand how you could sort out which numbers had exactly two rows of divisors.  My older son noticed that numbers that were the product of two primes (to the first power) would have exactly four divisors.  We worked through a couple of examples of integers of this form and found that they did indeed have that property.   My younger son was able to explain why this was the case – thanks to Art of Problem Solving’s Introduction to Number Theory book!  The last thing we talked about in this part was whether or not numbers of this form comprised all of the solutions to Dave’s problem:

Next up we started searching for other potential solutions.  We started off down the path of looking at powers of primes – it turns out this was a lucky road to head down.  We looked at squares and fourth powers, but neither of these types of numbers seemed to solve the problem.  It did give us the idea to look at cubes of primes, though, and that showed us one more set of solutions:

The last task was to see why the two types of numbers that we’d found – products of exactly two primes, and a single prime cubed – formed a complete set of solutions to the problem.     This part of the talk has a little more theory in it, but I think the lesson here is important – how do we know that our solution is complete?   We talk about how you count the number of divisors, and then how that counting process could arrive at exactly 4 divisors.   Funny enough, the divisors of 4 play an important role in answering that question.   The fact that there are only two ways to multiply integers together to arrive at 4 (2×2 and 4×1) tells us that we have a complete solution to the problem.  Yay!

So, in a couple of weeks we went from a neat problem shared by Tracy Johnston Zager about the sum of the divisors to a neat problem shared by David Radcliffe about products of proper divisors of integers.  Of course this journey was helped by the fact that I’m going through a number theory book with my younger son (and have gone through the same book with my older son previously).  These are challenging problems for kids to think through for sure, but I think that kids will enjoy the challenge.  These problems also do a great job of building up number sense because as you work through them you are constantly thinking about factoring, multiplying, and integers that share certain types of properties.    I forget the exact phrase, but in my mind problems like these definitely belong in the “low entry point / high exit point” problems that the people who study math education seem to really like.

As always, I’m glad that people are sharing problems like these ones on Twitter!

Ceva’s Theorem – a neat example of ratios in geoemtry

This morning my older son and I worked through a great example problem in Art of Problem Solving’s Introduction to Geometry book.  By amazing luck the section is one of the sections that Art of Problem Solving highlights on their web page about the book, so feel free to check out problem 5.7 here (and don’t peek at the solution!):


We actually came across the problem yesterday, but I wanted to devote an entire day to it today because the clever use of ratios in this problem is so instructive.  I definitely didn’t want the mathematical beauty in this example to be lost because we had to rush through it.  Also, we’ve been away from fractions and ratios for a while so my guess was that it would take a full hour to go through the problem in detail.  It did.

In the middle of talking through the problem this morning I remembered that the proof of Ceva’s theorem also uses ratios in a clever way, and thought a fun follow up on the example from this morning would be walking through the proof of Ceva’s theorem tonight.  It really is amazing that you can prove this beautiful theorem with just the area formula for triangles and a clever use of ratios.

In the presentation below, I’m following the proof given in section 1.2 of Geometry Revisited, which is where I learned about the theorem back in high school (and since I was too lazy to take a new picture, check out C.D. Olds’s Continued Fractions book too!!):

Book 2

I started with the statement of the problem and showed how to get started on the proof by making some simple observations about areas of triangles.  Then we began looking at the neat ratio idea:

Since this idea about ratios really isn’t that intuitive I wanted to take a little break from the geometry to just get a better understanding of why ratios behave in this seemingly strange way.  My son had the nice idea to look at the relationship abstractly to see why it was true.  It is a little funny that the relationship is easier to see abstractly than with specific numbers.

In the last part of the proof we show that the product of the three ratios is equal to one.  In the first video we showed that the first ratio we were looking at was equal to the ratio of the areas of two triangles.  We apply the same argument for the remaining two ratios and find two other sets of triangles whose areas are in the ratio that we were looking at originally.

If you are careful with how you label these triangles (and I wasn’t) you see quickly that the product of the three ratios is equal to one.  If you aren’t careful it takes a little bit of extra time to see that all of the products cancel.

So, a really fun example.  To a kid learning geometry it probably seems pretty surprising that a concept from arithmetic is going to lead to such an impressive result in geometry.  I like that aspect of showing this proof, too, and I also like the reverse – namely that a day of studying geometry gave us a great chance to review ratios.

A super fun day overall, and all from one cool example from our Geometry book!

The balance between “answer getting” and “mathematical thinking”

So – two similar posts on twitter this morning just a few minutes apart:

David Coffey here:


Keith Devlin here:

I struggled with the balance between answer getting and mathematical thinking just this morning with my older son.  I had him working through an example problem in the “angle-angle similarity” section from Art of Problem Solving’s Geometry book.   This is a topic that he’d not seen before, but he seemed to be able to work through the first couple of sample problems reasonably well.  The problem we recorded gave him a little more difficulty, though – particularly when it came to identifying how two similar triangles would match up.

I’m not satisfied with how I interacted with him in this problem and that dissatisfaction has been nagging at me all morning.  I’m not sure what specific thing I should have done better, but I definitely wish I had found a better balance between the answer getting part of this problem and the math ideas.

Terry Tao shares a math circle problem

I saw Patrick Honner tweet about Terry Tao’s math circle post last night:

Hopefully the link in the tweet to the original post works, but in case it doesn’t Tao’s post on google+ is here:

and here’s the problem itself:
“Three farmers were selling chickens at the local market.  One farmer had 10 chickens to sell, another had 16 chickens to sell, and the last had 26 chickens to sell.  In order not to compete with each other, they agreed to all sell their chickens at the same price.  But by lunchtime, they decided that sales were not going so well, and they all decided to lower their prices to the same lower price point.  By the end of the day, they had sold all their chickens.  It turned out that they all collected the same amount of money, $35, from the day’s chicken sales.  What was the price of the chickens before lunchtime and after lunchtime?”

In his post Tao asks not for the solution but for the thought process you went through to solve the problem.  I like these “thought process” posts, so I thought I’d give another one a shot.  My first one was here:

For this math circle problem I made the same assumptions that many of the commentators on Tao’s blog were making -> (i) the number of chickens sold by each of the farmers in both the morning and the afternoon was not zero, and (ii) that the both the morning and afternoon prices were an integer number of cents.

The first thing I did was look for a simple example just to get my arms around the problem.  With a morning price of $4 per chicken and an afternoon price of $1 per chicken you can get close to the situation the problem describes.  I didn’t put much thought into those two prices – I just chose those two prices because $4 was above the average price for the 10 chickens and $1 was below the average price of the 26 chickens.

Farmer 1:  8 chickens at $4 plus 2 chickens at $1 = $34
Farmer 2:  6 chickens at $4 plus 10 chickens at $1 = $34

Farmer 3:  3 chickens at $4 plus 23 chickens at $1 = $35

This short exercise gave me some hope that this initial guess was pretty close to the answer to the problem.

Next I wrote down some equations.   Call the morning price x and the afternoon price y, and let A, B, and C represent the number of chickens sold at the morning price by Farmers 1,2, and 3 respectively.  We have (in cents):

(1) Ax + (10 – A)y = 3500

(2) Bx + (16 – B)y = 3500

(3) Cx + (26 – C)y = 3500

A little equation combining, namely (1) + (2) – (3), yields the equation:

(4) (A + B – C)*(x – y) = 3500

Since both terms on the left hand side are integers the problem now is to find the right factors of 3500.  Fortunately 3500 doesn’t have too many factors, and double fortunately the example from the beginning tells me to expect (A + B – C) to have a value around 11.

There are two factors of 3500 that are on either side of 11, namely 10 and 14.  The values of (x – y) for those two choices are 350 and 250.  I checked 350 first.

Rewriting equation (3) you can get:

(5) C*(x – y) + 26y = 3500.

When (x – y) is 350 this equation becomes 350C + 26y = 3500, or

(6)  26y = 350*(10 – C).

This equation will not have any solutions when both y and C are positive integers since the left hand side is divisible by 13 and the right hand side isn’t.

Bad luck, I suppose, but we do have the second potential solution from above when (A + B – C) = 14 and the price change from morning to afternoon is 250 cents.  As Tao has asked to not give away the solution to the problem, I won’t work through that math but will say you can find a way to make both sides of the equation similar to (6) be divisible by 13 here.  Yay!

Definitely a fun challenge problem.  Unlike the Tim Gowers’s IMO problem that inspired my first “thought process” post, I hope to use this problem for a neat little Family Math project with my kids later this week!