Crazy Tangles and the order of the months

who: J1, J2, J3
when; a couple of minutes scattered through the day, 5 minutes at dinner
what did we use: two colored pieces of string (shoelace weight)

---

Playing with string

2 years ago, Fawn Nguyen wrote a note about Conway's Rational Tangles. It was a great activity for her algebra class (so go read her note) and I thought our little ones might enjoy it. While getting a soldering iron to fix a toothbrush robot (another upcoming post?) I found two blue and red string that looked really perfect for this investigation.

There are two moves, twist and rotate. Since Fawn does such a fantastic job of showing these moves, I won't even bother trying to describe them. For us, we made it into a two person activity where we hold different strings in each hand. I introduced the ideas in the following sequence:

1. told J1 that there were two moves and showed him. 
2. Then we did a bunch of twists and I counted one each time. 
3. After that, I told him a twist counts as +1. 
4. We did a sequence (3 twists, rotate, one twist, rotate, two twists, rotate, two twists)
5. at that point, he insisted on leading the moves, so he was in charge

J2 had been watching and basically understood the pattern, but we did something similar later as his turn.

J3 had her turn as well and basically just wanted to get tangled up in the string

Reactions
J1 got very excited about making conjectures about what the rotation operation does. His first guess was multiply by -1. We tested with twist, rotate, twist and found that we were back to state 0. However, when we did 2 twists, rotate, then 2 more twists definitely didn't get us to state 0.
Next conjecture was that rotate subtracts 2, but that also is disproven by the fact that TTRTT(0) is not 0. He is still working on it.

For J2, he liked to get complicated and started out with 10 twists and then a rotate. From that point, he let me call the moves (TRTTRTTRTTRTTRTTRTTRTTRTTRTT) and was totally delighted to see us get untangled back at state 0 after all of that.

J3 got a lot of mileage out of disrupting other people's tangles. This was certainly fun for her, but a bit frustrating for the older two.

Which month is first?

At dinner, I asked the kids which is first, January or February? J1 and J2 quickly said "January."
Next question, which is first, January or March? January again.
Next, January or April? Again, quick answer of January. P interjects, is that going to last?
January or May? J2 again says "January" quickly, J1 is starting to hesitate.
January or July? January again from J2, July from J1.

Why?
J2 is pretty clear; "January is the first month of the year, so it always comes first."
J1 has a different idea. It is June now, so our next July will come before the next January. Thus, July is first.

Next question, which comes first, January or December?
J2 again says "January, it is the first month of the year, so it has to come first."

What month comes immediately before January?
J2 says "December."

Note :J3 doesn't really know the months yet, so she wasn't very proactive, but she did enjoy repeating the others' answers.

This question occurred to me when reading Who's the Oldest over at Musings of a Mathematical Mom.