Starring at the Grid! (Continued)

[Sean Corey]

While starring at a sheet of grid paper on my plane ride home from the greatest workshop of all time (That's right, the AIM "How to Run a Teacher Circle" one), I came up wit a new grid game, slightly inspired by Conway's "Angels and Devils" Game.  I call it "3-Wall"

3-Wall

Mark any 2 lattice points A, B on a square grid. Find their taxi-cab distance.  Where should three line segments, each of length equal to the taxi-cab distance, be positioned to create the maximum obstruction from A to B?

Other variations, could be that the line segments are of length equal to the maximum horizontal or vertical distance between A and B. If obstructions must only be horizontal or vertical, only a few types of obstructions are useful, U, Z, |, L. 

Anyone have some other grid ideas, or comments on 3-wall?

Regards,

Sean Corey

[Joshua Zucker]

This really reminds me of the excellent board game Quoridor!  You take turns moving your piece toward the opponent's side of the board and then placing a wall to obstruct. Of course the problem is that the walls you place can also be in your way depending on how the game goes!  You try to give them a long path but you're not allowed to completely wall them in.  But you can lure them down a long pathway and then close it off at the last moment and make them retrace their steps.

I've played that a bunch with my kids, but not with my students.  It was nice with my kids, because we could play that I lost one wall every time I won (so the next game I could place fewer walls) until eventually hitting on the handicap that made it a fair game when I played as hard as I could.  I have a lot more fun playing games that way, where I can use every bit of strategy I have rather than holding back to let them win sometimes.

Angels and Devils is also an interesting one; it took a long time for it to be solved in general on the infinite board.  But you can play it on a smaller board, too, on your grid paper: the angel starts in the center and gets to move some number of spaces in any direction (for instance, like 2 king moves) and then the devils eat a square and try to prevent the angel from getting off the end of the graph paper by building a wall of eaten squares.

--Josh

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[Sean Corey]

I've never heard of Quoridor, but I am definitely going to check it out now!  

I have attached the start of a document I'm putting together.  It begins with the "apple orchard" problem, which makes me think about our grid explorations.  I saw the problem in Exeter's collection, then added some questions to it.  

Have any suggestions to add to these explorations?  I need some good problem solving questions that link to the conceptual/ syntax oriented material.  

-Sean