Terrible title, I’m sorry, but there is some truth in it, isn’t there? There’s a lot about mathematics that most people haven’t even experienced, even for those who do admit to liking or even loving it. Maybe I should be more specific here – I wish more people knew what mathematics can be. I’m not talking about learning mathematics (although it’s related, it’s not my focal point).

What I’m talking about is being mathematical, thinking mathematically, and doing mathematics.

“Problem Solving” is the gold standard of mathematics in education. It’s given praise and priority, and rightly so; it’s an extremely important and worthwhile way to use mathematics. Placing so much emphasis on solving problems, however, can create quite a one-dimensional view of mathematics. Mathematics is not just about solving problems, it is and can be so much more. Search “beauty of mathematics” on Google and you will be flooded with images, videos, and articles about the Golden Ratio, the Fibonacci Sequence, Pythagoras Theorem, and other fascinating things – but that is also not what I’m talking about.

What I’m talking about is being mathematical, thinking mathematically, and doing mathematics.

If it’s not problem solving or truly unbelievable mathematical relationships, then what is it? In my opinion, it’s the exploration of intuition. It’s feeling like you’re on the cusp of making a discovery or finding truth. It’s play, adding and removing parameters, or bending something until it breaks. It’s testing a guess and embracing the suspense of figuring it out. It’s wanting to show someone what you found – because it’s so cool!

But it’s not cool.

At least not to most people. Talking about mathematics is one of the few methods that will so reliably cause a person to roll their eyes. Why? Maybe because so many people haven’t experienced it for themselves. They might not know what mathematics can be beyond what they experienced in school. Mathematics is often so tightly tied to achievement and intelligence, most people might not believe they’re smart enough to even talk about it, which doesn’t make people feel great. When people don’t feel great, that’s not cool. So I get it. But…

Mathematics is not just needed in an exam.

Mathematics is not just printed in a textbook.

Mathematics is not just rearranging equations.

Mathematics is not just problem solving.

What I’m talking about is being mathematical, thinking mathematically, and doing mathematics.

Exploring your intuition starts by being comfortable to have an intuitive response. This is where problem solving can be an enticing avenue – like a crisis makes an interesting story plot, but that’s not the road I want to head down for now. Where I want to venture is play. Moments where your mind goes into overdrive pondering “what if” questions and playing out different scenarios. It’s when your mind is seemingly taken over by an idea or a question you need to find out more about. Someone else may have also had the same thoughts as you, in fact it’s almost certain that they have, but that doesn’t matter, because for whatever reason, it’s captured your attention – that’s mathematics. Exploring your intuition is forming and testing a conjecture.

Let me show you an example of this based on one of my experiences. I mentioned this a little while ago in a previous blog post, Normalising Not Knowing – The Balance Bike of Mathematics, and although it is a problem, that’s not my focal point here. Solving the problem was not the goal, the goal was to find something interesting to play with along the way.

12321 is called a palindrome because it reads the same backwards as forwards. A friend of mine claims that all palindromes with four digits are exactly divisible by 11. Are they?

Thinking Mathematically 2nd Ed. (Mason, Burton & Stacey, 2010)

If you haven’t tried it yourself, give it a go – you will find your own experience much more interesting than mine because it’s yours. If you solve it, ponder some “what if” questions. If something else grabs your attention while you’re solving it, follow that thought instead and dig deeper. Follow your nose. Hopefully it’s a fruitful endeavor and you learn something new that you wonder if someone else found out too.

What I love most about this problem is that it invites you to use your intuition. Intuitively, you’d think that all four-digit palindromes are divisible by 11 or that there exists a set of four-digit palindromes that aren’t and it’s your task to uncover them. But it’s not. Your task is to play. Use it as kindling for sparking something much more interesting than the answer.

Here’s my latest experience with it:

First, I divided the palindrome given in the problem by 11, which gave me 1120.9090909090… 12321 is a five-digit number. Idiot. But, knowing I probably wasn’t alone, I chuckled. The problem is playing with me, instead of the other way around.

Next I tried a few four-digit palindromes.

1221/11 = 111

5665/11 = 515

1001/11 = 91

2992/11 = 272

7777/11 = 707

9119/11 = 829

Feeling pretty convinced it’d always be true, my intuition took me to wonder about 6-digit palindromes.

123321/11 = 11211

“Wait, what?” I thought to myself, “123321/11 equals another palindrome!” Hooked by this thought, I tried another 6-digit palindrome…

321123/11

Which equalled 29119… 3! Damn. So close! Deflated, I did one more just to make sure 123321 was just an anomaly.

567765/11 = 51615

Whoa! Maybe it’s just palindromes that have digits that increase then decrease?

789987/11 = 71817 🙌

246642/11 = 22422 🤯

135531/11 = 12321 🤯 🤯 🤯

147741/11 = 13431 🤩

What about crazy big numbers?

123456789987654321 = 112233454332211

🤯 🤯 🤯 🤯 🤯 🤯 🤯 🤯 🤯 🤯 

There was no way I was putting this thing down. My intuition led me to form a conjecture. My intuition led me to bend my conjecture until it broke, adjust the parameters of the conjecture, and then bend it some more. My intuition led me to play.

From here, I honestly wasn’t sure how to best dig deeper. I was sitting on a plane with no pen or paper and I was doing operations using my laptop’s calculator, recording notes in a google doc. A visual model of what was going on and how it might explain what I was seeing wasn’t immediately obvious to me. So, I went to a very shorthand version of long division. My goal here was to dig into the smaller details to see if anything came out as I sifted out the big objects of the numbers going in and coming out of the calculator.

What I found was so mind-blowing, I couldn’t even really believe it.

I selected a palindrome that represented my conjecture with enough meat to surface something, but not too complicated that it was tricky to see. I wasn’t sure what I was expecting to find, but I was hopeful that this might make it a bit easier to notice a pattern. It felt like I was on the cusp of making a discovery.

123321 / 11

Like long division, I took each digit of 123321 working left to right to see if it was divisible by 11, joining the remainder to the next digit. Because it was just in a google doc, it looked weird and was a bit confusing to start with, but it was workable enough. I found that bolding the digits of 123321 made it easier to see what was going on.

1/11 = 0r1 → 12/11 = 1r1 → 13/11 = 1r2 → 23/11 = 2r1 → 12/11 = 1r1 → 11/11 = 1r0

Which shows that 123321 / 11 = 11211

There was nothing immediately jumping out at me here. To be honest, I was mostly surprised I did it correctly. So I tried another one.

357753 / 11

3/11 = 0r3 → 35/11 = 3r2 → 27/11 = 2r5 → 57/11 = 5r2 → 25/11 = 2r3 → 33/11 = 3r0

I was very proud of myself that I was able to develop a method of long division shorthand that worked, but it also highlighted a pattern I’m not sure I would’ve noticed if I didn’t do it this strange and confusing way. There was a pattern in the result of each operation.

3/11 = 0r3 → 35/11 = 3r2 → 27/11 = 2r5 → 57/11 = 5r2 → 25/11 = 2r3 → 33/11 = 3r0

0r3, 3r2, 2r5, 5r2, 2r3, 3r0

ANOTHER PALINDROME!

A common feeling now, but no less thrilling, I felt the need to try out another example to see if it worked.

159951 / 11

1/11 = 0r1 → 15/11 = 1r4 → 49/11 = 4r5 → 59/11 = 5r4 → 45/11 = 4r1 → 11/11 = 1r0

0r1, 1r4, 4r5, 5r4, 4r1, 1r0

WHOA! THEY ALSO FORM A TRAIN!

Each remainder equaled the whole part of the result in the next operation. I suspected that this might only hold true for the neat little examples that also fit my conjecture of increasing then decreasing digit palindromes. But hey, what did I really know? So I tried an example I was expecting to not work.

321123 / 11

3/11 = 0r3 → 32/11 = 2r10 → 101/11 = 9r2 → 21/11 = 1r10 → 102/11 = 9r3 → 33/11 = 3r0

0r3, 2r10, 9r2, 1r10, 9r3, 3r0

What this example showed me was how the remainder of 10 threw a spanner in the works. When written out like this, it made me suspect that this was what was causing a non-palindrome from being born by division of 11.

While I was thinking about why exactly a remainder of 10 might occur, my intuition led me to an idea that was too interesting to ignore. Was it possible to write out a random train of remainders and work backwards to recreate the starting number? So, I made up my own remainder train using random values that still fit the pattern I saw earlier.

0r2, 2r5, 5r3, 3r5, 5r2, 2r0

____ = 0r2 → ____ = 2r5 → ____ = 5r3 → ____ = 3r5 → ____ = 5r2 → ____ = 2r0

I started at the rear carriage of the train and worked through to the front (from the right side to the left side). I found that I already had a lot of information based on the remainder train I created. The first one (furthest right) looked like this:

____ = 0r2 → ____ = 2r5 → ____ = 5r3 → ____ = 3r5 → ____ = 5r2 → ____ = 2r0

22/11 = 2r0

The “22” was was determined by multiplying 11 x 2 and adding the remainder of 0. I then did this for the rest of the train.

2/11 = 0r2 → 27/11 = 2r5 → 58/11 = 5r3 → 38/11 = 3r5 → 57/11 = 5r2 → 22/11 = 2r0

278872 / 11 = 25352

Un. Be. Liev. A. Ble.

This was all too much for me at this stage. At every turn there was something new to be discovered and investigated. I never actually solved the problem given. I wasn’t even solving any problem at all. I was merely exploring my intuition. Some might say I posed my own problems to solve, but isn’t that just play?

This was a fun experience I’ll never feel comfortable mentioning to anyone because it’s not cool.

Now that’s a problem I’d love to solve.