# Dividing up land

Historical records indicate that the brothers Geoff, Petro, and Olaf divided up their large rectangular estate between themselves. Unfortunately, evenly dividing up favorable landmarks resulted in the complicated map below.

Each region here is owned by exactly one brother, the total area of each brother's regions are the same, and no brother owns two bordering regions.

Once the land was divided, one region remained which contained the main house. As all three brothers lived in that property, it has distinct borders with any other regions (So it can border any region, regardless of owner).

You have been tasked with finding the remains of that main house. You only have the unlabeled map provided.

To keep answers consistent, you can assume Geoff owned the land in the top right corner and Petro the region just to the left of that.

• In the above, I tried to come up with an original type of puzzle based on the knowledge that any map can be colored with exactly 4 colors. There are, therefore, 3 brothers and 1 capital. I randomly generated a much larger puzzle and modified it so it was solvable. I then made this little one so I could confirm if I had any fatal flaws with my design. I would love to see any loopholes you may find that I did not come up with. I expect this is a simple one for you all but it is reminiscent of Sudoku or minesweeper which people find fun. Feb 24, 2017 at 15:32
• Hey Kaine! Definitely a super fun puzzle. I thought the area condition was going to be a deal-breaker, but now I see that it's only utilized at the very very last bit. What you could have done instead is blended those last 4 squares into the large area surrounding them, and then in my opinion you get rid of the one pesky aspect of this otherwise really awesome puzzle! Feb 24, 2017 at 15:57
• @TheGreatEscaper Yeah, I did that several times to modify how the puzzle could be solved. Left that one to see the reaction. That says a lot. Basically, I need to write software for this a make very big ones, make them prettier, and remove any equal area requirements. As is this is a little too quick and easy. Putting them in books would actually help that. Good thing that finding the house did not confuse too much. That is another I thought I might have to remove. Feb 24, 2017 at 16:04
• There is at least one additional solution for the location of the house if we remove the equal area assumption. Feb 24, 2017 at 22:01

KEY: Red/blue/green represent the three owners.
The two shades of brown represent hypothetical deductions, where the two browns represent two owners in some order.
Grey represents a candidate for the house.

Looking at the red/browns, we see that if all those ares are owned by a single brother, we get a contradiction. So, one of those squares is the house.

Now we work from the other end. The browns represent red/green, but you don't know which is which until you do all this stuff which reveals that dark brown is red.

That gives you all of this, and you do some more hypothetical browns work to make deductions.

More hypothetical browns work (this time the browns are blue and green)

Note that the grid is 11x17 which is 1 mod 3, so the house takes up an area of 1 mod 3. This removes some grey possibilities. Also, I noticed that I coloured too many areas grey - the contradiction will still occur with a smaller subset of lands. So, we get this:

Now, use a quick hypothetical to see exactly which house will cause a contradiction:

So we know that little area is the house. Fill out the rest of the top left:

And you're done! Someone with patience can work out what the top right corner is supposed to be to satisfy the area condition :P

• The total land area is only needed to divide up the top right 4 squares. Feb 24, 2017 at 15:50
• Ah, silly me, there are some steps I didn't see :) Feb 24, 2017 at 15:52
• I don't mind one way or the other. What you've done here so quickly and your feedback tells me a lot about how the puzzle looks to those trying to solve. I really appreciate what you've done. Feb 24, 2017 at 15:57
• here is the picture which is a valid coloring but contradicts your 2nd and 3rd images, if that helps. yellow/orange is red/green or vice versa. purple/pink as well. Feb 24, 2017 at 17:04
• @Elias oops, that's a mistake on my part. I started off by colouring the 1x1 area to the upper left of the blue 'loch ness' shaped piece light brown (one of red or green), and forgot that it could actually still be blue... Feb 25, 2017 at 3:46