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[easy] Pieces in a Jigsaw puzzle

 
David O'Meara
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We've been getting puzzles out of the library and find that a 500 piece puzzle is a comfortable size for us. The problem is that not all of the puzzles have the number of pieces listed and the only reliable way to tel is to get all of the edge pieces, construct the two sides and multiply.

It occurs to me that it should be possible to give a reasoonable estimate of the number of pices given just the number of edge picees. I have a solution but I'll hold on in case anyone else is interested in looking. I bring this up now since the last jigsaw ddn't have the number of pieces listed, I have applied my logic to the edge pieces and after we put them together I'll be able to verify...
 
Ryan McGuire
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ASSUMPTION: To be most aesthetically pleasing, the proportions of the puzzle are 1:The Golden Mean or 1:1.618.

This gives an area of 1.618 sq units. In a 500 piece puzzle, it would take anout 309 pieces to cover 1 square unit. This means that a side 1 unit long has sqrt(309) or about 17.58 pieces pieces.

If that's rounded up to 18...
500/18 = 27.7someodd. Round that to 28. (18*28 is a 504 piece puzzle, so the buyer gets a four piece bonus.) 18+18+28+28 = 92. Since corners cound for two pieces worth of perimieter, that means that should be 4 corner pieces plus 84 other edge pieces.

If that's rounded down to 17...
500/17 = 29.someodd. Round that to 30. (17*30 is a 510 piece puzzle, so
the buyer gets a ten piece bonus.) 17+17+30+30 = 94. Since corners cound for two pieces worth of perimieter, that means that should be 4 corner pieces plus 86 other edge pieces.

Which is closest to the Golden Mean, 18:28 or 17:30?
28/18 ~= 1.55
30/17 ~= 1.76

1.55 is closer to 1.618 than 1.76, so I would guess that an average 500 piece puzzle is 18 by 28, yielding 4 corners and 84 other edge pieces.

(I could have gone with a 17*29 piece puzzle, but that isn't as close to a 1.618 proportion as 18*28 AND it cheats the buyer out of 7 pieces.)

How many total pieces and how many edge pieces were in the library puzzles, David?
[ July 31, 2006: Message edited by: Ryan McGuire ]
 
David O'Meara
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Basically my logic, but you answered the wrong question

There are 70 edge pieces in the jigsaw (including corners), what is your guess for the dimensions and number of pieces?
 
Ryan McGuire
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Originally posted by David O'Meara:
Basically my logic, but you answered the wrong question

There are 70 edge pieces in the jigsaw (including corners), what is your guess for the dimensions and number of pieces?


Ok, let's stick with the 1.618:1 ratio. If there are a total of 70 edge pieces, then the the perimeter is 74. Therefore the length and width add up to 74/2=37. 23:14 would be a ratio of 1.643. 22:15 yields 1.467. I would guess that the puzzle is 23*14 for a total of 322 pieces.

Am I close?
 
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