Tag Archives: Difficulty 8

Generalized Instant Insanity

The last problem in the “Games and Puzzles” section comes from a commercial puzzle (though one based on a much older puzzle).  It’s cool how these sorts of things pop up- a toy or game comes out and then people analyze it mathematically and find some neat features about it.

The problem: Generalized Instant Insanity.  This is problem GP15 in the appendix.

The description: Given a set Q of cubes, and a set C of colors we paint on the sides of each cube (and where |Q| = |C|) is there a way to arrange the cubes in a stack such that each color in C appears exactly once on each side?

Example: The Wikipedia page for the Instant Insanity game has a good description of the game and an example of the solution when Q = 4.

Reduction: Roberson and Munro reduce from “Exact Cover”.  I don’t think I’ve done this problem, but it’s basically X3C where the sizes of the sets can be any size.  (So it’s a trivial reduction from X3C).

They start by designing a graph representation of the puzzle, similar to the one shown on the Wikipedia page. Vertices correspond to colors, an edge connects two vertices if they are on opposite sides of the came cube, and the edges are labeled with the name of the cube.  The puzzle has a solution if and only if we can find two edge disjoint cycles that touch each vertex and edge label exactly once.  The two cycles correspond to the two sets of opposite faces we can see (since 2 sides of each cube are hidden by the stack).  Again, go look at the graphs in the Wikipedia article– they show this pretty nicely.

So, to perform the reduction, we’re given a set S {s1..sm} and a collection T of subsets of S, and need to make a graph (which corresponds to a puzzle).  Each element in S will become a vertex in the graph.  Each of these vertices will have a self-loop and be labeled ζi (So each vertex is a different color, and each self-loop is part of a different cube- so each cube has one color on opposite sides).

Each element si inside a set Th of T will also have a vertex: vi,h. This vertex has an edge with the next largest level sj within Th, wrapping around if we’re at the last element of Th. These edges are labeled ϒh,j.  We also have an edge from sjto vh,j if sj is in Th labeled with δh,j.  We also have 2 copies of self-loops from each vh,j to itself labeled ϒh,j.

They then add some extra edges to give the graph the properties:

  • One of the self-loops from vh,i to itself has to be in one of the two cycle sets.
  • If the edge from vh,i to sj (labeled ϒh,j) is in the other cycle set, then the edge from sj to vh,j (labeled δh,j) is also in that cycle set
  • The ζi edge label has to be used, and so will have to be used on an si vertex the cycle that uses that vertex will also have to have the labels ϒh,j and δh,j for some h.
  • The loops (labeled ζj and ϒh,j) are in one cycle set and the paths (labeled ϒh,j and δh,j) are in the other.

With these properties, then solving the puzzle means we have to find h1 through hm that correspond to sets Th in T, but also where there is a path ϒh,jh,j in the graph for each sj in Th_i.  We make sure no elements in S are repeated by making sure the puzzle only uses each sj vertex once.

Difficulty: 8.  I glossed over a lot of the construction, especially the definition of a “p-selector”, which is a subgraph that helps set up a lot of the properties above.  That definition is very hard for me to follow.

Left-Right Hackenbush for Redwood Furniture

Here’s an interesting problem, but a hard one to explain.  Most of what I’m doing here comes from the very cool “Winning Ways for Your Mathematical Plays” book by Berlekamp, Conway, and Guy, which I hope at some point in the future to have the time to really dig deeply into.  But for now, I’ll just use it as a reference to this week’s problem.

The problem: Left-Right Hackenbush for Redwood Furniture.  This is problem GP12 in the appendix.

The description: Ok, here we go.  First, a Hackenbush problem consists of an undirected, connected graph.  The edges of the graph are marked as “Left” or “Right”(though the book has some very nice colored pictures, where the edges are labeled “Blue” and “Red”).  Some of the vertices are on the ground (In G&J’s definition, there is one ground vertex, but it’s equivalent to having several vertices that are all on the ground).

On Left’s turn, they remove a blue edge and then all edges that are not connected to the ground are removed.  On Right’s turn, they remove a red edge, and then all edges that are not connected to the ground are removed.  A player loses if there are no remaining edges of their color.

redwood furniture Hackenbush instance is one where:

  • No red edges touch the ground
  • Each blue edge (or “foot”) has one end on the ground and the other touches a unique red edge (the “leg”)

Here are some redwood furniture instances from the book:

The “value” of a Hackenbush position is the number of “spare” moves (with optimal play) one player has after the other player loses.  A value of 0 means that whoever’s turn it is will lose (on an empty board).  The definition can be extended to fractional values.  For example, a value of 1/2 for (say) Left means that if we made two copies of the game, we would end up with a situation with a value of 1 for Left.

So, the question is, for some Redwood Furniture graph, and some K, is the value <= 2-K?

Reduction:

I’m just going to sketch the process here since it takes several pages of the book (and depends on results and ideas from earlier in the book).

They show:

  • The value of any redwood furniture graph is 2-N for some N.  In the degenerate case, the graph with just one Left edge has a value of 1. (=20)
  • On Left’s turn, they will always remove a foot (by definition, that’s all they have).  On Right’s turn, they should make a move that does not disconnect the graph, if possible.
  • A “Bed” is a redwood furniture graph where every red edge that is not a leg connects to a leg.  It has value 2-(m+1), where m is the largest number of moves that do not disconnect the bed.
  • The value of the game depends on how many extra moves Red has to get down to just a bed.
  • To find m, you need to know the size of the largest redwood tree (a tree is a graph that will be disconnected by the removal of any edge) that contains all of the legs.
  • The edges of the bed (the red edges that are not legs) form a bipartite graph.  So finding m is equivalent to the set covering problem, where the elements of the set are the vertices, and the edges are the sets.

Here’s how I think the Set Covering reduction works.  Given a set covering instance: a set S of elements, and a collection C of subsets of S, we’ll build a bipartite graph.  One set of the bipartite graph will correspond to the elements of S (and will be on the legs of the furniture graph).  The other set will correspond to the elements in C (and will be the vertices in the bed that are not in the legs).  An edge will go from each “C vertex” to each “S vertex” in its set.  Now, the cover is the set of vertices from the bed that cover all of the legs.

The book says you want the “smallest redwood tree which contains all of the legs”, which I think is the same thing (smallest number of extra vertices), but I’m not 100% confident since the Hackenbush game involves removing edges, and we’re choosing vertices in the cover.

I’m a little sad that the book does such a great job describing the game, and the value function, and then glosses over the reduction part (and uses misleading terms like “Minimum Spanning Tree of a Bipartite Graph”, which is a polynomial problem).  The actual reduction in G&J is to a private communication, so that’s not much help.

Difficulty: Boy, I don’t know, I think it depends on where you start from.  If my Set Cover reduction is the right one, and all you ask a student to do is that, it’s probably a 4.  If you’re going to make them prove all of the things I glossed over about the value number, then it probably goes up to at least an 8.

NxN Checkers

Back from my trip with a simple problem to explain, but a hard reduction to do.

The problem: NxN checkers.  This is problem GP10 in the appendix.

The description: Given a position on an NxN checkerboard, does black have a forced win?  It turns out the reduction will also work if we restrict the board to only having kings on the board (and so no “un-kinged” pieces)

Example: The “NxN” requirement is there since on a standard 8×8 checkerboard, there is a finite set of moves, and so theoretically you could solve the problem in O(1) time (for a really large constant factor, of course).  The starting configuration adds extra rows and columns of pieces to the board, still leaving two blank rows in between the two pieces.

So, let’s do an example on a 4×4 board.  The starting configuration is this:

* *
O O

(the dashes are empty spaces, * is Black, O is White)

Here is a configuration of pieces that will lead to a black win:

*
O
*

If it’s Black’s turn, they should move the piece in the second row up to either location on the first row (recall that all pieces are kings).  Then White’s only move is to go to the space Black just vacated, where it will be jumped, giving Black the win.

Reduction:

The paper by Fraenkel, Garey, Johnson, Schaefer, and Yesha contains a pretty detailed description of the reduction, which contains lots of complicated structures.  I’ll just give the general idea here.

The reduction is going to be from Geography, which is still NP-Complete even if the graph is bipartite and planar.  They create several structures to help them build their instance of the checkers game.

The first is what the call a phalanx– an open rectangle of (say) White kings that surround the (say) Black pieces.  The idea is that since there is no way for the Black pieces to jump anything in the rectangle, then White can “shrink” the phalanx towards Black, running them out of room to maneuver.  Here is a picture of a small phalanx on a 6×6 board:

O O O
O O O
O X O
O O
O O
O O O

..notice that whatever Black does, they will be captured on their next turn.  This remains true no matter how many Black pieces are trapped inside the phalanx, and no matter how much open space is inside the phalanx (White can use their moves to shrink it over time).

The key to the reduction is to build a set of interlocking “potential” phalanxes- situations where a Black king may be able to escape the phalanx.  If it can, Black can jump White’s pieces and win, but if it can’t, the phalanx will engulf Black and they will lose.  The geography instance is placed in the center of these potential phalanxes in such a way that a Black king can “escape” the Geography instance if and only if Black can win the geography game.  The reason why the Geography graph had to be planar was so that we could directly represent the vertices in the graph as positions on the checkerboard.  The reason why the Geography graph had to be bipartite was so that edges going from the first vertex set to the second could be all Black pieces, but the edges going from the second set to the first could be all White pieces.

The game starts with black at the “start vertex” for the geography problem, and jumping a line of White checkers:

When a vertex has more than one possible exit, that leads to more than one possible set of checkers to jump for the other player:

(This is part of figure 10 from the paper.  Here, after White jumps down the chain of Black pieces, Black can choose the chain of White pieces to jump through.)

The construction takes advantage of the rule in checkers (which I was not aware of until I was in my twenties!) that if a player can make a jump, they must make a jump.  So as long as players can jump checkers along these chains (alternately, as long as they can follow edges in the geography graph), they will.  As soon as a player cannot make a jump they will be able to deal with the Black king that can either escape the phalanx structure (and win for Black) or trap it (and win for White).

This is the general idea of the reduction, there are a lot of details that I am glossing over.

Difficulty: 8.  This is a bit hard to see and very hard to come up with, and it’s very easy to get lost in the weeds of the details.  I do like the way that the “removal” of edges from the Geography problem is modeled by the actual removal of pieces from the checkerboard, though.

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