For this month’s meeting, we returned to explore some numbers we first encountered in CAMI back in 2015 (Happy Numbers and the Melancoil).
After some community building, we started our exploration with a notice/wonder of this diagram.
WHAT DO WE NOTICE?
- Numbers on the same level share a digit – if 2 #s are linked horizontally, they have at least 1 digit in common – the numbers are reversed
- I see multiples of 7 in this tree (7, 70, 49)
- 44 looks lonely – it is sitting by itself – many other #s look like they have a partner or a friend with the same digits, but reversed
- I noticed that it is a web and that the paths all end.
- It looks like a family tree
- No loops are formed – you couldn’t walk around – you’d have to walk back
- 1 and 44 are hanging on by a string – 44 if you flip it is still 44)
- 7 could switch to 70 and 10 and 100 could flip to 1
- 130, 31 and 13 all fit together but only 130 and 31 are on the same row
WHAT DO WE WONDER?
- Is there a pattern that we can hold on to that we can use to figure out what is going on?
- What makes 100 go to 68-86?
- What is making the movement in this? What is generating these paths?
- Are the black lines/number fixed in this way or is it just how they are oriented that way at this moment – is this a biased way of viewing these things?
- What is the purpose?
Then Mark shared a little information with us.
The numbers in this diagram and known as Happy Numbers. The numbers in this diagram are the happy numbers between 1 and 100.
One way to determine if a number is a happy number is to square each digit in the number and add those squares together. Then repeat the process. If you reach the number 1, then your starting number is a Happy Number.
For example, here is what it looks like starting with the number 44.
Since we ended up at 1, we know that 44 is a happy number. Also any of the other numbers we produced along the way are also happy numbers (32, 13, and 10) because starting with any of them also gets us to 1.
Next we looked at another representation of some Happy Numbers.
These are all the numbers between 1 and 200. The numbers highlighted in blue are happy numbers (you’ll recognize some of them from the first diagram). All the other numbers are nonhappy numbers, which we called Melancholy Numbers.
WHAT DO WE NOTICE?
- We are squaring and adding and getting a smaller number at the end – that gets my attention because squaring and adding usually make bigger number
- I was looking to see if these happy numbers are colored in any type of pattern – but the gaps between them are different in each row
- There is no happy numbers between 141 and 166 (that looks like the largest gap in this table) – how large can a gap be
- I notice 3 columns with no happy numbers – 5, 15, and 18
WHAT DO WE WONDER?
- If we arrange the 1-200 in a different way, would a more noticeable pattern come forth?
- What about the non happy numbers? What happens to them? Do the other numbers ‘go infinite’? Rather than going to 1?
Next, we focused our curiosity on posing some math questions we could explore.
Some questions focused on changing the representation to find a pattern (since we couldn’t see one in the two representations we looked at):
- If we arrange the 1-200 in a different way, would a more noticeable pattern come forth?
- What would this look like in Base-2? Would that reveal a pattern?
- Are there ways to three-dimensionally arrange this grid to see a pattern?
Some questions focused on understanding more about the structure of happy numbers:
- No happy numbers between 141 and 166 (that looks like the largest gap in this table) – how large can a gap be?
- Do happy numbers have other properties in common besides happiness?
- Are multiples of 7 more likely to be happy numbers than other numbers?
- Is there a largest happy number?
- Does every happy number have a parent? Like 44 – is there a number that gets you 44?
- Sometimes there are consecutive happy numbers. What’s that about?
- What’s up with 5? Are there any happy numbers that end in 5?
There was also an interest in non-happy numbers:
- What happens to the other numbers? Do the other numbers ‘go infinite’? Do they make loops?
Folks choose an initial question to explore and spent some time doing some independent exploration.
A Tool for Exploring
Sarah Lonberg-Lew created an applet for generating a path from any starting number to see if gets to 1 (making it a happy number) or if something else happens (making it a melancholy number).
Then folks broke themselves into groups, with about half of us choosing to explore happy numbers and the other half choosing to explore melancholy numbers.
Some of the Work from Our Group Explorations and Discoveries
Happy Numbers: Slow Down
Happy Numbers: Speed Up
Melancholy Numbers: Slow Down
Melancholy Numbers: Speed Up
As promised, we ended with more curiosity and questions than we started with. Mark shared a few more tools and resources to continue our explorations:
An interactive way to visualize happy and melancholy numbers
For example, here is the visualization for 44 (a happy number)
Here is the visualization for 73 (a melancholy number)
Two Numberphile Videos:
- A video on Happy Numbers: “7 and Happy Numbers”
- A video on Melancholy Numbers: “145 and the Melancoil”
Some handouts for other ways to introduce Happy/Melancholy Numbers:
- Student Handout from Math Equals Love (gets students to test the numbers 1-100 to determine which are happy and which are melancholy).
- Purposeful Practice: Happy Numbers from Mark Chubb
- Further Explorations of Happy and Melancholy Numbers – created by Sarah Lonberg-Lew, this applet allows you to explore:
- For happy numbers: the number of steps for a happy number to get to 1, the number of steps for happy numbers to get to 1 within a range
- For melancholy numbers: the number of degrees of separation from the melancoil (melancholy loop), the number of steps for melancholy numbers to get to to the loop within a range, to find the distribution of degrees of separation from the melancholy loop within a range of numbers
