# United States of Squares

FabianFnas
Posers and Puzzles 23 Jul '08 06:25
1. 23 Jul '08 06:25
I notice that some states in USA are perfectly rectangular, like the state of Colorado. I don't know if there are such states or countries elsewhere in the world.

But actually there is such! Or if I dreamt it all up, I don't know, but there is. I call the country United States of Squares, or USS for short.

This country is rectangular in its shape, and moreover it is a federation of states where all states are perfect squares. (!) All of the states is also an integer multiple of a mile. (I don't know how much a mile is in this country, but it is not very long, you don't have to take the sphereness of the Earth into account, I don't even know if USS can be found on Earth, perhaps on another hypothetical Planet of some kind.)

In this country there are 9 states, for simplicity I call them One-state, Four-state, Seven-state, Eight-state, Nine-state, Ten-state, Fourteen-state, Fifteen-state, and Eighteen-state. Their shape is squared with their respective sides 1, 4, 7, 8, 9, 10, 14, 15, an 18 miles respectively.

Your problem is to draw a map of this rectangular nation. If you can present the solution as a map, you are good.
For me to know that you have the map right, please present the states that have mutual borders with One-state.

If I am blurry somewhere, please ask me so I can rephrase or explain.

If you have a solution, please mail me your answer in order not to spoil the fun for the others. I will present who was first after a while.
2. 23 Jul '08 07:491 edit
Originally posted by FabianFnas
I notice that some states in USA are perfectly rectangular, like the state of Colorado. I don't know if there are such states or countries elsewhere in the world.

But actually there is such! Or if I dreamt it all up, I don't know, but there is. I call the country [b]United States of Squares
, or USS for short.

This country is rectangula ] in order not to spoil the fun for the others. I will present who was first after a while.[/b]
And I have the correct answer by sonhouse! Well done!

I'm glad he didn't spoil the problem for the rest of you, by presenting his answer in this thread.
3. forkedknight
Defend the Universe
23 Jul '08 14:20
Here's what I've deduced in the last few minutes:

The total area is 1056

Possible rectangles are (22,48), (24,44), and (32,33)

For the 22 length side, the only blocks that fit between 18-state and the side are 1-state and 4-state, so that doesn't work

For the 24 length side, there is no combinations of blocks that fit in the 18x6 area, so that doesn't work

That leaves just the 32x33 shaped USS
4. forkedknight
Defend the Universe
23 Jul '08 14:29
FYI, a spreadsheet is very handy for mapping this problem.
5. 23 Jul '08 14:31
Originally posted by forkedknight
Here's what I've deduced in the last few minutes:

The total area is 1056

Possible rectangles are (22,48), (24,44), and (32,33)

For the 22 length side, the only blocks that fit between 18-state and the side are 1-state and 4-state, so that doesn't work

For the 24 length side, there is no combinations of blocks that fit in the 18x6 area, so that doesn't work

That leaves just the 32x33 shaped USS
You mailed me the correct answer.
You're the second one coming up with it.
6. 23 Jul '08 15:02
Now, find ANOTHER rectangular country composed of 9 squares of different sizes (no 2 squares are equal).
Don't be too smart and just multiply all lengths in the original problem by a certain number - that is not, of course, a different solution.ðŸ™‚

A more difficult problem (it has a solution):
Find a SQUARE country composed of squares of different sizes.
7. 23 Jul '08 17:41
Solution to original problem submitted. Wish I could have gotten on sooner as the starting point has been mentioned already, but was not mentioned when I read it this morning.

Solution was found shortly after beginning to work it.
8. 23 Jul '08 18:57
Originally posted by geepamoogle
Solution to original problem submitted. Wish I could have gotten on sooner as the starting point has been mentioned already, but was not mentioned when I read it this morning.

Solution was found shortly after beginning to work it.
And you solved it nicely, geepamoogle! Well done!
Well done to mtthw too, who solved it correctly!
9. 23 Jul '08 20:06
Originally posted by David113
Now, find ANOTHER rectangular country composed of 9 squares of different sizes (no 2 squares are equal).
Don't be too smart and just multiply all lengths in the original problem by a certain number - that is not, of course, a different solution.ðŸ™‚

A more difficult problem (it has a solution):
Find a SQUARE country composed of squares of different sizes.
There is actually a USS that itself is a square. It has 24 states.