I occasionally find problems to solve or scroll back through my camera roll for screenshots of problems shared online. I use these problems not to keep my mathematical skills sharp, but to rekindle my joy for mathematics and the continuous wonder it provides my mind. Sometimes I get to do this through my work, and passion and joy get to intersect, but rarely do I find enough time to truly follow my nose – to get to, what I believe, is the essence of what it means to think and work mathematically… to explore my intuition.

I’ve written about this before, but the more I think about it, the more strongly I feel that there is not enough time and support for students to explore their intuition through mathematics at school. I sometimes wonder if debates about how students best learn mathematics eclipses how students do mathematics and I am left to question whether students ever do their own mathematics, or if it’s just always somebody else’s – answering questions that they didn’t ask and solving problems that have already been solved.

Let me try to explain what I mean with an example.
(as always, try it yourself first)

Consider two whole numbers (for example 3 & 6). These will be the first two numbers. The third number is the sum of the first two (9). The fourth is the sum of the previous two (15), and so on (3, 6, 9, 15, 24, 39, …). What do the first two numbers have to be such that the fifth number is 100?

Seed Numbers, featured on Peter Liljedahl’s website

This was a problem to solve for me. I didn’t know what to do, so I did my usual first step when I problem solve – I tried to make sense of it. I needed to know more about the problem, perhaps by finding a relationship or pattern that existed, something that might illuminate a path towards a solution. Since I had nothing to generalise from, I began specialising by trying some examples.

2 and 7
2, 7, 9, 16, 25, 41, 66

2 and 8
2, 8, 10, 18, 28, 36, 64

2 and 5
2, 5, 7, 12, 19, 31, 50

Hazaa! 50! Wait, I think I was supposed to get to 50 on my fifth number…
“What do the first two numbers have to be such that the fifth number is 100?”

You idiot. The goal is 100, not 50.

2 and 7
2, 7, 9, 16, 25, 41, 66, 107

2 and 8
2, 8, 10, 18, 28, 36, 64, 100
🎉🎉🎉

Wait, that’s my eighth number 🤦‍♂️
Wait again! It could be my fifth number, if I count back!

2 and 8
2, 8, 10, 18, 28, 36, 64, 100

Take that, Peter Liljedahl.

Ok, is 18 and 28 the only correct response? I bet it’s not.

At this moment, my time was cut short by the cry of a baby who was supposed to be asleep for another 7 hours. So, as I rocked her back to sleep in the dark, I fired up Desmos on my phone. Game on.

Here was the correct answer I stumbled upon by chance:

18 and 28
18, 28, 36, 64, 100

Setting up two sliders on Desmos, I could quickly test out values to see what ones added to 100 on the fifth number.

Wait, what? 120?? Oh you double idiot. 18 + 28 = 46, not 36 🤦‍♂️

So, it turned out I had no correct response, but with a baby still stirring in my arms and not ready to put down yet, I decided to venture onwards using my phone – brightness dimmed all the way down by the way #suchAGoodDad

Adjusting it to 8 and 28 seemed to do the trick.

8, 28, 36, 64, 100

Dragging the slider below 8 only seemed to decrease the fifth number below 100, and above 8 increased it above 100.

7 and 28
7, 28, 35, 63, 98

9 and 28
9, 28, 37, 65, 102

So I figured that 8 was the only solution for 28, but what about 29?

8 and 29
8, 29, 37, 66, 103

7 and 29
7, 29, 36, 65, 101

Ok, still a little too high. How about 6 and 29?

6, 29, 35, 64, 99

Ok, now it’s too low. 5 will only take it lower, but what if we increase 29 to 30?

Aha! 5 and 30
5, 30, 35, 65, 100

I noticed that the next one that worked decreased the first number by 3 and increased the second number by 2. 

8 and 28 ✅
5 and 30 ✅

So, my next attempt was…

2 and 32
2, 32, 34, 66, 100 💪

This would mean that -1 and 34 would be…

-1, 34, 33, 67, 100 🤯

So my solutions so far were:

8 and 28
5 and 30
2 and 32
-1 and 34

Graphing these out on my phone (as I rocked my daughter back to sleep) looked like this.

If negatives weren’t freaky enough, I thought I’d try to figure out what it would be if the first number was 0. According to the line that was forming on my graph, it should be…

0 and 33⅓
0, 33⅓, 33⅓, 66⅔, 100
 🤯🤯🤯

This all really got me thinking.. 33⅓ was quite significant as it’s ⅓ of 100. What if we were trying to get to 100 on the sixth number instead?

I adjusted my sliders and the first answer I found was 5 and 17

5, 17, 22, 39, 61, 100

Increasing the second number, I noticed that neither 18 or 19 worked.

4 and 18
4, 18, 22, 40, 62, 102

3 and 18
3, 18, 21, 49, 60, 99

2 and 19
2, 19, 21, 40, 61, 101

1 and 19
1, 19, 20, 39, 59, 98

Leaving 20, which was peculiar, because it only worked if I used 0 as the first number.

0 and 20
0, 20, 20, 40, 60, 100

This result also struck a chord with me – 0 and 33⅓ was a solution for the original problem, and 0 and 20 was a solution to this new one. In terms of 100, that’s ⅓ then ⅕. I was surprised it wasn’t 25, which would’ve been ¼, but this is how it turned out and I was interested to see what fraction might pop out if we wanted the seventh number to be 100.

On this occasion I was met with a little roadblock. Adjusting the slider for the second number led to the two following responses.

0 and 13
0, 13, 13, 26, 39, 65, 104

0 and 12
0, 12, 12, 24, 36, 60, 96

Which led me to try 0 and 12.5
0, 12.5, 12.5, 25, 37.5, 62.5, 100

12.5, which is ⅛ of 100, was also a bit surprising to me, and it wasn’t until I lined it up with the others that I realised what was going on…

⅓, ⅕, ⅛

I was so excited to test out my next guess, for when we wanted the eighth number to be 100.

Writing it out as fractions, however, lifts the veil.

0 and 100/13
0, 100/13, 100/13, 200/13, 300/13, 500/13, 800/13, 1300/13 = 100

As it does for the others:

0 and 12.5
0, 100/8, 100/8, 200/8, 300/8, 500/8, 800/8

0 and 20
0, 100/5, 100/5, 200/5, 300/5, 500/5 = 100

0 and 33⅓
0, 100/3, 100/3, 200/3, 300/3 = 100

The only thing left was to put my daughter in her cot and record my thoughts.

This was utter fun. What did I do? Sure, I solved the problem, well not really. I thought I did, but then realised I read the question wrong. Then, I thought I solved it again, but learned that I actually just read my answer wrong. Upon reflection, I think there were a couple key moments.

1. I didn’t pack it up at 8 and 28. A gripe I have with problem solving is that it’s too answer focussed. I don’t actually care too much if my answer is correct (or, evidently, if I’ve read the question correctly). I care more about the questions I ask myself along the way or after I’ve solved it. My main aim is to find something interesting, something I didn’t expect, something profound.

2. I explored my intuition and chased my curiosity. By exploring my intuition, I’m determining the path I take, following what piques my interest, and ultimately, finding joy. For me, joy is not finding out that my answer is correct when I check the back of the textbook. I find joy in the beginning – when I don’t know where my first few steps will take me. I find joy at the first plot twist, then the second, and if I’m lucky enough the third – when my intuition is proven wrong. I find joy at the end – when I look back and reflect.

Since leaving the classroom, perhaps my perspective of what it means to do mathematics has ventured even further away from the content-highway students and teachers are faced with in schools. I’m not suggesting that all problems lend themselves to such rich exploration, nor am I suggesting that this is the only way to do mathematics, but what I am suggesting is that all students get an opportunity to explore their intuition, chase their curiosity, and answer their own questions.