Powers of a permutation and Santorini review

Two unrelated topics today. Or ... I guess one could argue that everything in math is related, but I don't see direct connections myself. If you spot some, let me know in the comments.

Rainbow permutations

We have a collection of crayons that can stack. Well, to be honest, mostly the kids break the tips off. The second most common use is to stick them on fingers as fancy fingernails. The fourth most common use is to actually draw or color with them.

J2 was engaged in the third most common use, stacking, when he noticed something new. He started with the colors stacked in rainbow order (R O Y G B Purple Pink). Then, he took 2 off the bottom and moved them to the top. Then, he took the bottom two and moved to the top and repeated. He noticed that, eventually, he got back to the starting order. Experimenting further, he tried the same process moving 3 from the bottom and repeating. Again, he eventually got back to the starting order.

Here is an example of the crayons stacked together:

In this case, he is moving blocks of 5:

After showing me, he suggested trying 4, 5, 6, then 1. He noticed a couple of things:

• moving 6 is like moving 1 backwards (from the top to the bottom). 
• Similarly, 5 and 2 are related, 4 and 3
• 7 is prime, so maybe that is the reason the arrangement repeats

To test his hypothesis, he added another crayon (for 8) and tested. Again, he got back to the original arrangement. Hmm, doesn't need to be a prime!

Where should we take this first foray into group theory?

Santorini

Gordon Hamilton of Mathpickle, one of our favorite game and puzzle resources, has a Kickstarter for the newest version of his game, Santorini.

I encourage you to check it out. (For what it is usual disclaimer applies, I'm not financially related to this game or producers in any way.)

For reference, this is what the previous version of the game looks like:

Our DIY board (version 1)
While this version of the game will have nice custom pieces, the underlying game can be played with almost any collection of stackable objects. For our introduction to the game, we tried using our TRIO blocks:

To explain what you're seeing:

• 1x1x1 cubes are used for building levels. I originally thought we would color coordinate (red for first level, pink second, etc), but J1 and J2 liked mixing up colors.
• 4 unit straight connectors for our builders: magenta versus green.
• Angle or arc connectors are radio antenae to serve as the fourth building level and block further building
• Flags and wings as boundaries of the 5x5 board

Reactions

Both older Js enjoyed the game play. We quickly played 5 or 6 times. This game clearly has a lot of depth. We will have to play a lot more to see what patterns we can identify and whether we can develop any opening strategies. They are also very eager to play with some god powers. Perfect way to spend the rest of the holiday this week!

The TRIO blocks have pros and cons for this game. One of the best features is that it makes the game set-up robust to tipping the board, knocking the pieces off, or otherwise unsettling the position. It was clear to see the sizes of the levels and also immediate to identify towers that had already gotten killed (with a fourth level antenna).

In our play, there were two drawbacks. First, the straight connectors are a bit hard to remove from their positions. Second, the higher towers (2 and 3 levels) sometimes obscured allowed diagonal moves in a way that the stacking tiles version didn't. I wonder if this will also be the case with the new version of the game as the building levels seem tall relative to the size of the builders.

Longer-term, I wonder about buying customized games vs playing with generic materials. It is easy for the kids to grab a box off the shelf and start playing. Somehow, I suspect they will be less likely to grab a building set and a notecard with rules.

More thoughts on DIY game versions

Before Santorini, I had gotten excited about making DIY versions of the GIPF project games. I was thinking:

1. these games will be fun to play, so great motivation to replicate them
2. interesting challenge to make our own triangular grid board (squares we already have in abundance)
3. good creativity prompt as we repurpose items
4. it would make the kids feel power over the game structure and rules, leading to exploration of variations and deeper thinking about structure.

However, the kids, particularly J1, were surprisingly unenthusiastic. With Santorini, again, they didn't take much initiative in developing our DIY version. However, I wonder if they will be more motivated to modify or replace the TRIO blocks board since they now see that the game is fun and have experienced some of the limitations of our current version.

On the other hand, maybe they'll just push me to buy the commercial version.