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Math Help - Real-world dissection problem

  1. #1
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    Real-world dissection problem

    Too many years since plane geometry was easy for me to visualize!

    I have a sheet of material, exactly 3 feet by 5 feet = 15 sq ft.
    Can I make a few straight cuts and reassemble it into a 4 ft square with a 1 ft square hole in the center (15 sq ft, net)?

    Thanks for the help; my tile tabletop budget thanks you, as well!

    Dave
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  2. #2
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    Afterthought:

    I should clarify that request.

    There's the trivial solution, dicing the sheet to 1 ft squares, and the better one of cutting it into 4, 18x30" tiles. Is there a cut that would yield two "L" shaped pieces that would assemble to a square with a hole?

    What about assembling to a 48" square, with a 16" hole?

    Dave
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  3. #3
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    Quote Originally Posted by dbell View Post
    I have a sheet of material, exactly 3 feet by 5 feet = 15 sq ft.
    Can I make a few straight cuts and reassemble it into a 4 ft square with a 1 ft square hole in the center (15 sq ft, net)?
    5 cuts will do it.
    Get a piece of cardboard and cut out a 3 by 5 inches rectangle.

    Cut 1: cut out from one of the 3inch sides a 1 by 3 rectangle;
    so you're left with a 3 by 4 and a 1 by 3

    Draw a 4inch line on the 3 by 4 splitting it exactly in half; so the
    3 by 4 is now divided in two 1.5 by 4 rectangles (but not cut).

    Draw two 3inch lines, each 1.5 inches from the 3inch sides; so the
    3 by 4 now "shows" two 1 by 1.5 rectangles "down the center".

    Cuts 2,3,4: cut out one of these 1 by 1.5 rectangles.

    Slide down the one you cut out by 1 inch: that leaves a 1 by 1
    square exactly at center of the 3 by 4, and leaves the 1 by 1.5
    an inch "out" of the 3 by 4; and we see 2 empty 1 by 1.5 areas
    needing to ne filled to complete the 4 by 4.

    Cut 5: cut the 1 by 3 (from cut 1) in half: gives you two 1 by 1.5

    Place them to fill the 2 empty areas.

    Hope you were able to "follow" all that...IT WORKS!!
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  4. #4
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    Quote Originally Posted by Wilmer View Post
    Draw two 3inch lines, each 1.5 inches from the 3inch sides; so the
    3 by 4 now "shows" two 1 by 1.5 rectangles "down the center".

    Cuts 2,3,4: cut out one of these 1 by 1.5 rectangles.
    I did get lost in this pair of steps, but I see how it works out in the end.

    More cuts and small pieces than I was trying to achieve, but clever...

    I think I'll end up cutting th four 18"x30" pieces, and adding th extra framing bits, but I'll give it a day or so, to see if any brilliant insights show up.

    Thanks!

    Dave
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  5. #5
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    Quote Originally Posted by dbell View Post
    Too many years since plane geometry was easy for me to visualize!

    I have a sheet of material, exactly 3 feet by 5 feet = 15 sq ft.
    Can I make a few straight cuts and reassemble it into a 4 ft square with a 1 ft square hole in the center (15 sq ft, net)?

    Thanks for the help; my tile tabletop budget thanks you, as well!

    Dave
    You need 2 (= two) straight cuts, along the mid-parallels of the rectangle.

    Re-arrange the 4 congruent rectangles (see attachmant). Equal colours indicate equal rectangles.
    Attached Thumbnails Attached Thumbnails Real-world dissection problem-tischmitloch.png  
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  6. #6
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    Here comes a 3-cut-version - but I suspect that it is nothing but Wilmer's solution ...?

    (The cuts are painted in red)
    Attached Thumbnails Attached Thumbnails Real-world dissection problem-tisch3schnitt.png  
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  7. #7
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    Quote Originally Posted by earboth View Post
    You need 2 (= two) straight cuts, along the mid-parallels of the rectangle.

    Re-arrange the 4 congruent rectangles (see attachmant). Equal colours indicate equal rectangles.
    That's the (non-trivial) one I started with, and most likely what I'll end up using. This is a cutting pattern for tile backer board for a 4'x4' table top. I have a 3x5 sheet of board, and am putting a firebowl in the middle of the table. Before going out and buying a second sheet and butchering both to cut two 2'x4' pieces, I thought I'd put some effort into the problem...

    Thaks for the suggestions!

    Dave
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