December 15
The mathematics of Level 3
[Yes, this post is all about mathematics and patterns. If that sounds horrific, please feel free to skip it. If you want though, I've created a new Level 3 pattern for you to try at the bottom.]
Why would you write about this?
My handle is the Dancing Geek. The geek part is not just for show – I have a degree in this stuff, and sometimes I actually enjoy solving unnecessary puzzles just for fun.
The theory behind Level 3 as a mathematics puzzle
I’ve made up some notation to keep this shorthand. H is horizontal, V is vertical and the number relates to the position. So H1:V1 is left arm in Horizontal 1 position, right arm in vertical 1 position. Clear?
1 complete round (i.e. you get back to the starting position and are ready to go on the opposite side) is 16 moves at Level 3. It breaks down into 4 repetitions of the same 4 moves (i.e. the instructions are the same). Looking at where your arms are after each of these 4-move combos for the first starting position gives us the pattern:
H1:V1 to V1:H3 to H3:V3 to V3:H1 to H1:V1
This can be generalised (using algebra – yippee!) to:
Hx -> Vx
Vy -> Hy+2
In words:
A horizontal starting position will end with that arm in the same position on the vertical plane.
A vertical starting position will end with the arm in the transquartered position on the horizontal plane. (Hy+2 uses modulo 4 arithmetic, in case you weren’t clear).
How does it work?
Another way of looking at it would be to say that the movement swaps the positions of the arms, moves the right arm 2 positions on, and swaps the planes of the arms.
Either way, the most important aspect of the pattern is that it has symmetry 4. That is after 4 repeats it returns to where it started. Simple example of this would be a rectangle with a dot in the corner being turned 90 degrees clockwise:
Example of 4 symmetry
A brand new level 3 pattern!
Yes, I have created a new pattern which I am calling Level 3 because it works in a similar fashion. Since I don’t know about anything higher except for a bit of Level 4, I could be talking complete and utter bull. Blame @Sari_O for getting the idea stuck in my head.
It goes:
- Transquarters
- Mirror reflection
- One forward, one back
- Mirror reflection
Or, you could do:
- One forward, one back
- Transquarters
- One back, one forward
- Both forward
Not sure if I’ll try these or not.
Woo hoo! Thank you for totally making my brain go “huh?!?!” today! It took a few times before I got it right in the H1-V1 starting position. Obviously I had to try both of them as soon as I could.
What threw me a few times was that the individual cycles didn’t always end in 1-3 -positions, although the 4×4 cycle did end in the starting position H1-V1. Got over it, though, once I got the 4×4 done and was safely back where I started. I wonder if it was just me doing it wonky or if it happens for other people as well.
I have a feeling these will become a part of my mornings for the next few weeks…
Sari’s last blog post..Perfectionism, pessimism, and fear
@Sari – At least I know there’s one person out there giving it a go!
The only pattern that’s common to all three versions (the original, and my two made up ones) is the 4-symmetry, so the individual cycles won’t fall into the same places. The idea was just to come up with something that when repeated four times would get you back to the starting position, without getting there after fewer cycles of 4 (both the original and at least one of mine do go through the starting position during one of the individual cycles).
I would love to know what your experiences with these are!
oh my god!!!!
charles faris’s last blog post..friday afternoon update! 9: i’ve got a secret
“Looking at where your arms are after each of these 4-move combos for the first starting position gives us the pattern:
H1:V1 to V1:H3 to H3:V3 to V3:H1 to H1:V1″
Apparently I was so excited about the whole new patterns thing that I fluently assumed you were talking about your own sequence there. Duh. Sorry.
Talk about preconceptions making your life difficult.
It’s interesting to have another sequence for a level, though. With the first two, you get into the pattern and it automates, but with this level, you get two dimensions of difficult: the non-repeating formula itself and then the choice between which non-repeating formula you’re doing.
Case in point: in your first sequence, my arms want to continue forward after the transquarter, and I have to focus to not do the original pattern. Focus and mistakes. Brilliant.
Sari’s last blog post..Four Life Lessons from my sister
@Sari – re: making you have to go wrong and focus – I’m suddenly thinking that I really should be trying these out too at the moment given my struggle to make myself go fast enough to get flummoxed. Hmmm, I will try one today and see what happens.
I’m totally chuffed that you like them
[...] session was 10 minutes of my first alternate to Level 3 starting with the right side [...]