Thread: Modelling a puzzle mathematically?

How can I model this puzzle mathematically?
My friend had showed me this puzzle from the game Professor Layton and the Curious Village and I was wondering how it could be solved mathematically rather than using trial and error.

Here's the puzzle:
{
A man who lives in a red house, a man who lives in a yellow house, a man who lives in a blue house and a man who lives in a green house all have their houses built on the same land lot, and want to divide up the property evenly.

They live on a 6 by 6 grid, furthermore each house needs a well on its property.

The land also must be divided up such that the shape surrounding each person's land is the same.
}

Here's a visual representation (wells are grey) :
http://filesmelt.com/dl/squares.JPG

Thanks in advance! Sorry if I didn't make this very clear.

Originally Posted by m84uily

How can I model this puzzle mathematically?
My friend had showed me this puzzle from the game Professor Layton and the Curious Village and I was wondering how it could be solved mathematically rather than using trial and error.

Here's the puzzle:
{
A man who lives in a red house, a man who lives in a yellow house, a man who lives in a blue house and a man who lives in a green house all have their houses built on the same land lot, and want to divide up the property evenly.

They live on a 6 by 6 grid, furthermore each house needs a well on its property.

The land also must be divided up such that the shape surrounding each person's land is the same.
}

Here's a visual representation (wells are grey) :
http://filesmelt.com/dl/squares.JPG

Thanks in advance! Sorry if I didn't make this very clear.

Much of mathematics, particularly at the highest levels IS trial and error. You guess the right answer based in intuition, insight and trial and error and then prove that your guess is correct.

Does a house occupy a single square ? A well ?

One way would be the obvious 4 sets of 3x3 squares on the grid. This sounds too easy and I suspect that there are some constraints that you have omitted.

I see, thanks for explaining to me why trial and error isn't necessarily a "bad" way to go about solving things.

I'll try to clarify on the rules:

-When the land is divided up, each house must have its own piece of land and cannot be sharing its land with another house.
-Houses may not share their wells.
-The shapes created to contain the houses may not overlap.
-All houses and wells cannot be moved from their initial position
- houses and wells both occupy a single 1x1 unit square

red house is located in the 1st row 3rd column
yellow house is located in the 1st row 4th column
a well is located in the 2nd row 5th column
blue house is located in the 3rd row 3rd column
a well is located in the 3rd row 5th column
a well is located in the 6th row 2nd column
a well is located in the 6th row 5th column
green house is located in the 6th row 6th column

divide up the property evenly.

That part has a simple mathematical answer, 36/4=9.
I'm assuming that the puzzle is discreet, that individual squares may not be subdivided.

The land also must be divided up such that the shape surrounding each person's land is the same.

Did you mean "area" when you wrote "shape"? Unless the plots are not required to be contiguous it is impossible to create four translatable geometric figures on the template provided. If there is no translational requirement then the puzzle has multiple solutions. If there is a translational but not a contiguous requirement then I don't know. I am guessing that the area of math that deal's with this sort of thing is topology, maybe graph theory, possibly knots. I used colored pencils and graph paper to reach my conclusions.

You forgot about rotation. In this puzzle (I've beaten all three Layton games ) rotation is allowed.

One way to reduce the amount of trial and error is by thinking about partial solutions and what they imply. For example, if one house had all four of its surrounding squares in the same lot, all the houses would have to, which is obviously impossible, so you can rule that out. (In fact, the green house shows that each house can have at most two of the neighboring squares in the same lot.)

With rotation there is one unique solution. Don't know how to break it down mathematically. All I can say is it's "stringy". Where do I find the rest of the "Layton" puzzles?

Oh, I had thought a right angle triangle was still a right angle triangle when rotated 90 degrees from its initial position, sorry for the confusion!

http://professorlaytonwalkthrough.blogspot.com/

Here's a walk through for the Layton games, just don't look at the answers if you're interested in solving the puzzle on your own :)

All of the Professor Layton games are great games with great puzzles and a good story and cast, so if you have a DS, I'd recommend just buying them. If you don't have a DS, I guess it might not be worth getting one just for this series (not that there aren't a lot of other great DS games :wink.