The Corral Puzzle

I just finished a script on The Corral Puzzle.

I really enjoy puzzles. Even more than simply solving puzzles, I like understanding the thought process behind the puzzle process and trying to generate similar puzzles that are difficult in their own right yet still solvable. So far I’ve written works on The Sudoku Puzzle, The Nonogram Puzzle, and The Shade the Cells Puzzle. Today, I want to write about the Corral Puzzle.

The story behind how I discovered this puzzle is twofold. Initially, a coworker brought a Corral puzzle to my desk to see if I could solve it. It took me a minute, but I was able to get to the correct solution. We did a few more afterwards and I wrote some scripts to try to create new instances of these puzzles, but for a while it remained in the “unfinished work” category. Then sometime earlier this year, I purchased a book “Brain Workout: Math & Logic Puzzles” by Dave Tuller and Michael Rios. I was astonished to see that the first puzzle in this book was the Corral puzzle (which is where I was able to learn the name of the puzzles, as before they were just some things presented to me by a friend). The book has about 20 challenging puzzles (all of which I have not solved), but I was left again wondering how they came up with the puzzles. Before going further, I’ll give the rules of the puzzles and an example of a problem with the corresponding solution.

We are given a square grid (say 4 rows and 4 columns, or 5 rows and 5 columns, or in general n rows and n columns). Inside the grid some of the cells have a number inside of it. The object of the puzzle is for the user to draw a bag (also known as a corral) around the numbers inside the grid. The limitation is that the numbers tell how many neighboring cells can be “seen” from the given cell, looking only horizontally and vertically in both directions without reaching an endpoint of the bag. There are no restrictions on the shape of the bag except that it actually represents a closed loop inside the grid.

Consider the following example:

In this example, we are told the following hints:

  1. Cell (1, 2) (row 1, column 2) can see 6 of its neighbors.
  2. Cell (3, 1) can see 3 of its neighbors.
  3. Cell (4, 2) can see 5 of its neighbors.
  4. Cell (4, 3) can see 5 of its neighbors.

From these hints, we can reach the following solution:

Our corral in the solution would be to place the contiguous block of blue (turquoise) cells into a bag. We can check that the solution satisfies the assumptions as follows:

  1. Cell (1, 2) can see cells (1, 1), (1, 2), (1, 3), (2, 2), (3, 2), and (4, 2), which is 6 cells.
  2. Cell (3, 1) can see cells (3, 1), (3, 2), and (3, 3), which is 3 cells.
  3. Cell (4, 2) can see cells (1, 2), (2, 2), (3, 2), (4, 2), and (4, 3), which is 5 cells.
  4. Cell (4, 3) can see cells (1, 3), (2, 3), (3, 3), (4, 3), and (4, 2), which is 5 cells.

So we can see that this solution satisfies our assumptions.

Right now, I’m able to work with this by using it as an instance of The Set Cover Problem. The set that we would like to cover is the set of cells inside the Corral. The possible subsets are the cells in the corral that can be viewed by each cell inside the bag. Although Set Cover is an NP-Complete problem, we can still find feasible solutions using a number of algorithms. Here, I generate a feasible solution using the greedy approach.

There is still a problem with ambiguity. Sometimes the initial puzzle generated will allow for multiple Corrals to fit the original description. This is true with the example given as the following is also an optimal solution.

There is a thin line between revealing enough cells to ensure a unique solution and revealing too many cells such that the problem becomes trivial to solve. I’m still working on that and I may update the script in the future if I decide that the ambiguity is too much to live with.

So, enough talk. Go and check out my script on The Corral Puzzle and let me know what you think.

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3 thoughts on “The Corral Puzzle

  1. Hello,
    Your numbering system is DIFFERENT from the numbering system used in the Tuller/Rios book (and also elsewhere). In your system, if a cell has the number 7, it means that it can “see” 7 other cells.

    In the Tuller/Rios book, if a cell says 7, it means that it can see 6 other cells — that is, the number includes ITSELF as one of the cells. If you read the description in that book carefully, you will see what I mean.

    So if a cell says “n,” it means it can see “n-1″ other cells.

    In fact, I have found other online sources that uses the same numbering system as that book.

    I personally prefer the system that you use, but for whatever strange reason, the other system is actually more common. I just thought that I would let you know about it.

    Thanks.

  2. Hello,
    I had to post a second message. I went and checked your script and generated some puzzles. Interestingly enough, in your script, you have used the numbering system used in the above-mentioned Tuller/Rios book !

    In addition, there is a big problem with your solutions. The “corrals” that you are providing as solutions are NOT connected — that is the red cells are all NOT connected as one unbroken loop.

    But, the conventional format of corral puzzles is that the corral cells have to be connected as one unbroken loop. Again, check the Tuller/Rios book and you will see what I mean.

    Thanks.

    1. First, thank you for both your comments. In regards to your comments, two things:

      – I need to clarify the statement about the cells that can be seen from a given cell. You bring up a good point that needs to be clarified regarding whether the given cell is actually counted as a cell that is seen or not, so thanks for noticing that.

      – In regards to the second point though, the corral is the set of blue cells (not the red cells) which should always be a continuous block. As far as I remember, Corral Puzzles as described by the Tuller/Rios book doesn’t make any requirements on the cells not contained in the Corral. Again, this is a point of confusion that I should (and will re-word) to clarify. Thanks for bringing this to my attention.

      Hope you enjoy the script though.

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