How many squares will fit in a rectangle?

by sonone, May 18, 2020, 6:17 PM

Hi!
After the MATHCOUNTS target round on AoPS, this idea came to my head, a generalization of one of the problems: "what is the least number of squares (any size) that can fit in an $\it x\times y$ rectangle without extra space?" (note, if $x$ and $y$ are not integers, multiply both sides until you get integers). I tested allot of rectangles and ended up saying the answer is $$\left\lceil\dfrac{x+y}{2\gcd(x,y)}\right\rceil.$$
I would be grateful to anyone who could prove or disprove this.
This post has been edited 4 times. Last edited by sonone, May 20, 2020, 10:31 AM

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Can you specify? Do the squares need to be the same size? Are the side lengths integers?

by CaptainFlint, May 18, 2020, 7:00 PM

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I clarified. :)
This post has been edited 1 time. Last edited by sonone, May 18, 2020, 7:05 PM

by sonone, May 18, 2020, 7:04 PM

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How can you define greatest common divisor if $x$ and $y$ aren't integers? Would you just scale the rectangle until one of the side lengths was an integer?

by SirCalcsALot, May 19, 2020, 8:01 PM

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You would have to multiply both sides by some other number. If the rectangle is 1.3 by 4.1, you multiply by 10 to get a 13 by 41 rectangle with has the same ratio of dimensions as the 1.3 by 4.1 rectangle.

by sonone, May 20, 2020, 10:56 AM

Old material is mostly Asymptote, new material is calculator programming

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  • I still exist as well.

    by G.G.Otto, Aug 11, 2023, 2:44 AM

  • hello I'm still here lol

    by player01, Aug 6, 2022, 6:24 PM

  • [REVIVAL] I will start posting more calculator relating posts very soon. Even though school has been busy, I have been programming my calculators a decent amount, so I have a lot to share...

    by sonone, Feb 18, 2022, 10:29 PM

  • wow its been like 2.5 years since geo class

    by pieMax2713, Feb 4, 2022, 8:38 PM

  • @violin21, I've been very busy with school lately and haven't been able to add another lesson. I will when i get a free moment

    by sonone, Aug 19, 2021, 12:45 AM

  • ORZ CODER

    by samrocksnature, Aug 9, 2021, 9:57 PM

  • Could you make more Asymptote lessons on your "How to do Asymptote" blog?

    by violin21, Aug 9, 2021, 7:26 PM

  • You can take it, just C&P the CSS into your CSS area

    by sonone, Apr 17, 2021, 10:08 PM

  • how can we take the CSS if we have permission to not take it?

    by GoogleNebula, Apr 17, 2021, 5:22 PM

  • That is awesome!

    by sonone, Apr 15, 2021, 10:09 PM

  • I modified your dodecahedron and got:
    [asy]
    import three;
    import solids;
    size(300);
    currentprojection=orthographic(0,1.3,1.2);
    light(0,5,10);

    real phi=(sqrt(6)+1)/3;
    real g=(phi-1)/2;
    real s=1/2;
    real a=sqrt(1-phi*phi/4-g*g)+phi/2;

    triple[] d;
    d[0]=(phi

    by Andrew2019, Mar 26, 2021, 12:15 AM

  • Not too many, just changing the color here and there. I really like your CSS!

    by sonone, Feb 2, 2021, 10:35 AM

  • Nice!

    I see you're making changes to the CSS. :)

    by G.G.Otto, Feb 1, 2021, 9:26 PM

  • I'm learning Java now!

    by sonone, Feb 1, 2021, 5:56 PM

  • And I took part of it from CaptainFlint and then added a ton of modifications. ;)

    by G.G.Otto, Dec 1, 2020, 8:56 AM

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