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  • 8+8=5+13 ? Why

    I have a mathematical puzzle.
    8 x 8 inch square, cut into four pieces, becomes a 5 x 13 inch triangle.
    I have shown this puzzle to numerous people, no one has the answer.
    I am unable to insert image.
    If anyone could help me with this and explain the difference in the mathematics I would appreciate it.

    Lampman
    PS. I can e-mail a picture.

  • #2
    Re: 8+8=5+13 ? Why

    **************************************************
    * !!! Spoiler Warning !!! Spoiler Warning !!! Spoiler Warning !!!
    *
    * Do not read this post if you want to solve the puzzle on your own
    *
    * If you want to solve the puzzle on your own do not read this post
    *
    * !!! Spoiler Warning !!! Spoiler Warning !!! Spoiler Warning !!!
    **************************************************


    Hello Lampman,

    This is a very nice puzzle, and I remember being pleasantly baffled when I first encountered it in the past.

    I haven't seen your picture of the puzzle, but I'm almost certain that it will be basically the same as my attached diagram.

    The puzzle is based on the fact that drawings can sometimes be slightly deceptive, because it's hard for them to capture the same precision that numbers do.

    In my diagram, the pieces A through D on the left are arranged into an 8x8 square with an area of 64 square inches. On the right, the same pieces are arranged into a 5x13 rectangle of area 65 square inches. So where did the extra square inch come from?

    The answer is that the pieces on the right do NOT fit together perfectly, and there is a small gap between the upper and lower halves, with an area of 1 square inch.

    One way to demonstrate the existence of a gap is to focus on the point I've labeled 'X'. If we treat the point labeled 'O' as the origin of an x-y coordinate system, then piece C has a vertex (corner) at (5,2), meaning 5 inches to the right and 2 inches up from O.

    But what are the exact coordinates of the point X opposite this vertex, on the hypotenuse of the triangle B? First, note that the slope of the hypotenuse is equal to "rise-over-run," or 3/8, because it is 3 inches high and 8 inches wide. Point X is at a horizontal coordinate of x=5. Therefore, its vertical coordinate is y=5(3/8)=15/8. So point X has coordinates (5,15/8), or 1/8 inch below the point (5,2) where piece C has its vertex! This means that there's a tiny gap of 1/8 inch there, which is hardly noticeable on a diagram.

    It may be possible to demonstrate this puzzle physically by cutting pieces of cardboard or wood to the exact shapes in the diagram, but it would require some care to do so with the necessary precision.

    Neat, huh?
    Attached Files
    Last edited by ; 12-09-2007, 09:21 AM. Reason: Orthography

    Comment


    • #3
      Re: 8+8=5+13 ? Why

      Ron

      I am a machinist, I have cut it in metal exact.
      It is a true 8 x 8, and a true 5 x 13, to the thousands of an inch.
      We will wait for more feedback.

      Lampman

      Comment


      • #4
        Re: 8+8=5+13 ? Why

        Terrific! It would be great to have an actual experimental test.

        When you lay out the pieces in a 5 13 rectangle, you'll need to make sure that the opposite sides of the rectangle are precisely straight and parallel, and that the corners are precise right angles (90). Otherwise, the pieces could end up pushed together so that there's no gap in the middle, but then the outer perimeter would no longer be a true rectangle.

        Comment


        • #5
          Re: 8+8=5+13 ? Why

          Attached is an image thats been floating around the internet. Its not the same as your puzzle, just something that reminded me of it.

          I haven't tried to verify this, just saved it when I came across it.
          Attached Files

          Comment


          • #6
            Re: 8+8=5+13 ? Why

            Roy

            Your attached thumbnail is a correct representation of the puzzle.
            Since I said I was a machinist, the measurements are exactly as I stated.
            Anyone can test it for themselves.
            I appreciate the interest, and curiosity.
            With the possibility that I'm right, my question is, (HELP) how can it be?.
            Lampman

            Comment


            • #7
              Re: 8+8=5+13 ? Why

              Hi Lampman,

              Thank you for the follow-up. Yes, it is really strange, isn't it? And yet, I want you to realize that everything in the puzzle is completely logical and follows all the rules of mathematics. There's no monkey business going on, as I hope you'll convince yourself.

              Here's the single most important thing to notice. In the 5 13 rectangle of my diagram, the diagonal sloping line formed by the top edges of pieces B and D is not really a single straight line. Instead, it is two separate lines--piece D has a slightly steeper slope on top than piece B, and this creates the gap between the top and bottom halves of the rectangle.

              I know the result seems impossible, but that's why it's a good puzzle! And thank you to you for asking about such an interesting problem, and for using your skill to show that this puzzle really is true.

              Can any other readers help explain to Lampman why this surprising result is true?


              Originally posted by Robert Fogt
              Attached is an image thats been floating around the internet. Its not the same as your puzzle, just something that reminded me of it.

              I haven't tried to verify this, just saved it when I came across it.
              Thanks Robert, your posted diagram is based on the same principle as mine--namely, that the hypotenuse of the overall triangle appears to be a straight line, but in fact it is not.

              The hypotenuse of the red triangle has a slope of 3/8 or 0.375. The hypotenuse of the green triangle has a different slope of 2/5 or 0.400. So the outline of the overall shape is not even really a triangle--its "hypotenuse" consists of two line segments with different slopes. There's a slight break in the slopes at the juncture of the red and green triangles, but it's too small for most people to notice.

              The red and green triangles in the top diagram are arranged so the overall "hypotenuse" is slightly concave and encloses a smaller area for the overall shape. In the bottom diagram, the positions of the red and green triangles are reversed to form a slightly convex "hypotenuse" that creates enough space for one additional unit of area (the "hole").

              The same basic technique is used in the original puzzle. Looking at my diagram from Post #2, the hypotenuse of triangle B has slope 3/8 or 0.375, while the top edge of trapezoid D on the right has slope 2/5 or 0.400. The sloping "line" formed by the top edges of pieces B and D in the 5 13 rectangle is not a single straight line at all, but two segments of differing slope, with a break at the juncture between pieces B and D!

              Comment


              • #8
                Re: 8+8=5+13 ? Why

                Originally posted by lampman
                Roy

                Your attached thumbnail is a correct representation of the puzzle.
                Since I said I was a machinist, the measurements are exactly as I stated.
                Anyone can test it for themselves.
                I appreciate the interest, and curiosity.
                With the possibility that I'm right, my question is, (HELP) how can it be?.
                Lampman
                Hi Lampman, your 5 x 13 rectangle includes the area of your pieces, plus the area of the small gap that Roy has marked with the letter "x".

                The area of the gap must therefore be exactly 1. Hard to credit, as it is long and skinny, so it looks like almost nothing.

                In practical terms, you can see this making sense when you build a house and make a hall 100mm (4"?) wider, which looks like nothing on the plans. It doesn't make the rooms at the end of the hall much bigger, but costs heaps more, and makes the total floor area quite a bit bigger. - In fact, you think to yourself, "Why didn't we just buy a different table that would have fitted down the hall more easily???"

                Comment


                • #9
                  Re: 8+8=5+13 ? Why

                  Roy types faster than me

                  Comment


                  • #10
                    Re: 8+8=5+13 ? Why

                    Lampman,

                    Try making a square 16 x 16 or 24 x 24 etc... and doing the same practical experiment.

                    Maybe the increase in size will help you see the "gap" or help you measure the angles more precisely.

                    Thanks Mrs. X. That is a very good illustration.

                    As I navigate a boat, I assure you that a 1 degree mis-alignment is quite costly in both fuel and time.

                    Comment


                    • #11
                      Re: 8+8=5+13 ? Why

                      Hello Lampman,

                      Here's yet another explanation that might help. Sometimes a problem is easier to understand after considering a more obvious example.

                      So, have a look at the diagram attached to this post. On the left-hand side, I've drawn four identical pieces of size 3" 1", placed next to each other in a rectangle with a total area of 3" 4" = 12 square inches.

                      On the right-hand side, I've rearranged the pieces into a square of size 4" 4" = 16 square inches. So the area has gone from 12 to 16 square inches, a magical increase of 4 square inches!

                      But wait a minute, you're saying to yourself, that's ridiculous, the extra 4 square inches come from the empty space inside the square, you can't count that as part of the area of the pieces.

                      Well, the same thing is happening in the original problem. The 5 13 rectangle includes the long, narrow empty gap of 1 square inch. That doesn't count as part of the area of the metal pieces. No metal was magically created, you're just enclosing some empty space that shouldn't be counted.

                      Although the 5 13 rectangle occupies a total of 65 square inches, that total is made up of 64 square inches of metal, and 1 square inch of empty space.
                      Attached Files

                      Comment


                      • #12
                        Re: 8+8=5+13 ? Why

                        FOR GEOMETRY BUFFS AND MATH HOUNDS

                        In the past, I had simply accepted on faith that the empty gap has an area of 1 square inch in the rectangle shown in the attachment to Post #2. Now I've actually verified that fact, and here are some pointers for any students or readers who want some hints on doing the same.

                        The gap is actually a very squashed and tilted parallelogram, whose area is obtained by the same simple formula as for a rectangle, area = base * height, or A = b*h:
                        Parallelogram - diagram & formulas
                        http://z.about.com/d/math/1/0/A/F/parallelogramr.gif
                        The base of the parallelogram (which is also the hypotenuse of triangle B) is given by the Pythagorean theorem as
                        b = sqrt(82 + 32) = sqrt(73)
                        For the height of the parallelogram, note first that the vertical separation between point X and the adjacent vertex of trapezoid C is 1/8 (see explanation in Post #2).

                        Next, you'll need to convince yourself that the height of the parallelogram is therefore 1/8 times the cosine of the narrow angle of triangle B at point O. But that cosine is the ratio of "adjacent over hypotenuse" of triangle B, or 8/sqrt(73), and the height of the parallelogram is therefore:
                        h = (1/8)*(8/sqrt(73)) = 1/sqrt(73)
                        Finally, the area of the parallelogram is ...
                        A = b*h = sqrt(73) * 1/sqrt(73) = 1
                        I geometry!

                        EXTRA CREDIT. How many degrees is the very narrow angle of the gap between pieces B and C at point O in the diagram in Post #2?

                        Answer: 90 - arctan(5/2) - arctan(3/8) = 90 - 68.2 - 20.6 = 1.2

                        This shows how careful you'd have to be in cutting out physical pieces to demonstrate the puzzle for yourself. A combined error of only 1.2 in the angles at which you cut or place the pieces would completely destroy the effect.

                        It was great to have a precision machinist like Lampman take an interest in this puzzle!

                        Comment

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