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Thread: [SOLVED] Help with this puzzle?

  1. #1

    [SOLVED] Help with this puzzle?

    Hey guys i need to solve this puzzle. Anyone know the solution? I have attached the picture which i tried to recreate as best as i could in paint and the question is as follows: How many quadrangles in the picture are there???[IMG]file:///C:/DOCUME%7E1/Owner/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG][IMG]file:///C:/DOCUME%7E1/Owner/LOCALS%7E1/Temp/moz-screenshot-1.jpg[/IMG]
    Attached Thumbnails Attached Thumbnails [SOLVED] Help with this puzzle?-untitled.jpg  
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  2. #2
    Sep 2007
    According to Wikipedia and Google a 'Quadrangle' is basically a 'Quadrilateral' or four-sided shape. So I suppose the answer is 18?

    - However, if these shapes were meant to be joined together then that would be a different matter (seems 'too' easy a question to have them separate).
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  3. #3
    Sep 2007

    One approach

    I would approach this by considering pairs of rectangles. Start in the lower left corner, and call that rectangle your "base" rectangle. Then consider each of the rectangles (including the base rectangle) and answer, "Do the outermost corners of this rectangle and my base rectangle form the corners of a valid quadrangle?"

    Write the total in the base rectangle. (I make it 19 for the rectangle in the lower left, if I've interpreted your diagram correctly.)

    Iterate by repeating with each of the other rectangles as base rectangle, one after the other in a compact pattern, but considering only those quadrangles that do NOT include a rectangle with a number already written in it. When done, add up the numbers in each of the 18 rectangles.

    Another approach is to construct the set of all possible pairs of rectangles (there are 18 rectangles, so there are 18 x 17 / 2 = 153 possible pairs), and eliminate those pairs whose outer corners don't define a valid quadrangle. Add the 18 single rectangles and eliminate duplicates where the four corners of the quadrangle are in 3 or 4 separate rectangles, and you should get the same total.
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  4. #4
    Super Member

    May 2006
    Lexington, MA (USA)
    Hello, benjihag69!

    I see no method except Brute Force counting.

    How many quadrangles are there?
          |   A   |   |   D   |
          *-------* C *-------*
          |   B   |   |   E   |
          |   F   | G |   H   |
          | I | J | K | L | M |
          |   |   |   |   |   |
          | N | O | P | Q | R |
          |   |   |   |   |   |

    I started in the upper-left with $\displaystyle A$ and moved downward,
    . . then I moved across and down.

    $\displaystyle A,\;AB,\;ABF,\;ABFIJ,\;ABFIJNO$
    . . $\displaystyle ABC,\;ABCFG,\;ABCFGIJK,
    . . $\displaystyle ABCDE,\;ABC{D}EFGH,\;ABC{D}EFGHIJKLM,$ $\displaystyle ABC{D}EFGHIJKLMNOPQR$

    Then start with $\displaystyle B$ and move down and across.

    $\displaystyle B,\;BF,\;BFIJ,\;BFIJNO,$

    And proceed with each block in the same way.

    $\displaystyle C,\;CG,\;CGK,\;CGKP,$
    . . $\displaystyle CDE,\;CDEGH,\;CDEGHKLM,\;CDEGHKLMPQR$

    $\displaystyle D,\;DE,\;DEH,\;DEHLM,\;DEHLMQR$

    $\displaystyle E,\;EH,\;EHLM,\;EHLMQR$

    $\displaystyle F,\;FIJ,\;FIJNO,$
    . . $\displaystyle FG,\;FGIJK,\;FGIJKNOP,$
    . . $\displaystyle FGH,\;FGHIJKLM,\;FGHIJKLMNOPQR$

    $\displaystyle G,\;GK,\;GKP,$
    . . $\displaystyle GH,\;GHKLM$
    . . $\displaystyle GHKLMPQR$

    $\displaystyle H,\;HLM,\;HLMQR$

    $\displaystyle I,\;IN,\;IJ,\,IJNO,\;IJK,\;IJKNOP,\;IJKL,\;IJKLNOP Q,$
    . . $\displaystyle IJKLM\;IJKLMNOPQR$

    $\displaystyle J,\;JO,\;JK,\;JKOP,\;JKL,\;JKLOPQ,\;JKLM,\;JKLMOPQ R$

    $\displaystyle K,\;KP,\;KLPQ,\;KLM,\;KLMPQR$

    $\displaystyle L,\;LQ,\;LMQR$

    $\displaystyle M,\;MR$

    $\displaystyle N,\;NP,\;NOP,\;NOPQ,\;NOPQR$

    $\displaystyle O,\;OP,\;OPQ,\,OPQR$

    $\displaystyle P,\;PQ,\;PQR$

    $\displaystyle Q,\;QR$

    $\displaystyle R$

    I'll let you count them up . . .

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  5. #5
    Sep 2007
    I see that we’ve all assumed the shapes are joined and fit into a 5 x 5 sized area (considering the smallest side of a quadrangle as 1 unit) as it looks like this was the intention.

    Soroban described an excellent way of systematically working through each possibility. The reason this is so effective is that your not going to miss any if you go through every option, every possible quadrangle of a 5 x 5 grid (unless you skip or miscount).

    I make it 97.

    - - - - - - - - - -
    Soroban missed 'LM' but other then that they've also got 97.
    Last edited by Obsidantion; Sep 17th 2007 at 12:13 PM.
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