Squares

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DanishDynamite
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Location: Copenhagen

Squares

Post by DanishDynamite »

Prove that for any whole number N > 5, a square can be divided into N squares.

(I suspect this is the type of problem where you either see the answer fairly quickly or it takes you hours to see the solution. I was clearly in the last category.)
gnome
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Location: New Port Richey, FL

Re: Squares

Post by gnome »

DanishDynamite wrote:Prove that for any whole number N > 5, a square can be divided into N squares.

(I suspect this is the type of problem where you either see the answer fairly quickly or it takes you hours to see the solution. I was clearly in the last category.)
Ok... for even N's, take a square from the lower right corner and adjust its size so that the remaining space can be filled with an odd number of squares. The larger the corner square, the more small squares can fill the rest.

For odd N's... cut out a large square in the center of the bottom edge. Adjust size to need as with the even solution.

Hardly rigorous, but I think it makes it clear enough.
DanishDynamite
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Joined: Mon Jun 07, 2004 4:58 pm
Location: Copenhagen

Re: Squares

Post by DanishDynamite »

gnome wrote:
DanishDynamite wrote:Prove that for any whole number N > 5, a square can be divided into N squares.

(I suspect this is the type of problem where you either see the answer fairly quickly or it takes you hours to see the solution. I was clearly in the last category.)
Ok... for even N's, take a square from the lower right corner and adjust its size so that the remaining space can be filled with an odd number of squares. The larger the corner square, the more small squares can fill the rest.

For odd N's... cut out a large square in the center of the bottom edge. Adjust size to need as with the even solution.

Hardly rigorous, but I think it makes it clear enough.
"Hardly rigorous" seems right. :D

I can understand your solution for even N's as it is the one I found, but your solution for odd N's leaves me puzzled.

Could you post a diagram?
gnome
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Joined: Tue Jun 29, 2004 12:40 am
Location: New Port Richey, FL

Re: Squares

Post by gnome »

DanishDynamite wrote:
gnome wrote:Ok... for even N's, take a square from the lower right corner and adjust its size so that the remaining space can be filled with an odd number of squares. The larger the corner square, the more small squares can fill the rest.

For odd N's... cut out a large square in the center of the bottom edge. Adjust size to need as with the even solution.

Hardly rigorous, but I think it makes it clear enough.
"Hardly rigorous" seems right. :D

I can understand your solution for even N's as it is the one I found, but your solution for odd N's leaves me puzzled.

Could you post a diagram?
...

Actually I'm incorrect, you still get an even number. Let me work on it some more.
ceptimus
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Location: UK

Post by ceptimus »

Trying to hide this in a spoiler ('select' it, or Ctrl-A to view)

<table bgcolor=white><tr><td>
Given any existing arrangement, any existing square can be subdivided into four equal ones, yielding an extra three squares in total. So if we can solve for any three consecutive numbers, then all numbers greater than that are proven soluble by induction.
<pre>+-+-+-+<br>| | | |<br>+-+-+-+<br>| | | = 6 (and therefore 9, 12, 15...)<br>| +-+<br>| | |<br>+---+-+
+-+-+---+<br>| | | |<br>+-+-+ |<br>| | | |<br>+-+-+---+ = 7 (and therefore 10, 13, 16...)<br>| | |<br>| | |<br>| | |<br>+---+---+
+-+-+-+-+<br>| | | | |<br>+-+-+-+-+<br>| | |<br>| +-+ = 8 (and therefore 11, 14, 17...)<br>| | |<br>| +-+<br>| | |<br>+-----+-+<br></pre>
</td></tr></table>
DanishDynamite
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Joined: Mon Jun 07, 2004 4:58 pm
Location: Copenhagen

Post by DanishDynamite »

The cepter is a must, for ceptimus!

A different inductive proof than mine, but just as valid. Well done!

(My proof was based on dividing the case into even and odd numbers of squares. The bit which took me forever to see was how to make a division into 7 squares.)
gnome
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Post by gnome »

Beautiful. I was approaching that, but I don't think I could have made the argument so elegantly.

:clap: