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Jim T.

Puzzled

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Okay, this is the duplicate that the other two pictures of the puzzle pieces are laying on.

I can understand the physical easier than the theoretical. Either it fits or it doesn't.

BTW Thank you Afet. I followed your instructions and this was much easier.

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The point being is that without computing it myself, I agree with Snowyh's approach.  I do believe that the solution is mathematical and not illusional.  I just haven't figured out the formula. 

Actually, what I was trying to offer in my earlier post is the mathematical proof, stated from a slightly different perspective than Roz's explanation, that the puzzle is indeed an optical illusion. The key is--neither composite figure is a perfect triangle. (They're close, which is why your rough-cut carpentry failed to reveal the illusion.) If they were, then the sum of the areas of each internal piece would equal the calculation of the area of the composite (length x height / 2). But they don't. Fig 1's area is a little short, and Fig 2's area is a little over what they should be--hence the concave/convex discussion. The difference is very small, which is why the illusion is so convincing!

This was a good puzzle. Thanks for posting it.

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This has been fun. Thank all of you for joining in with me.

From a purely mechanical standpoint, I see no illusion Pieces 1,2,3 and 4 fit the solid piece in one configuration and not the other. They were both initially cut identical. Mirror twins - so-to-speak. Then I cut one apart using the graphs meaurements. Allowing for sloppiness of saw and tool cuts, they still don't add up to one square inch. If I were to put the sawdust back in somehow, they would fit exactly one way, but not another.

Yes - I'm baffled, but that is the fun of it.

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I still think the sweater drawer analogy is a good one :P, but as most of us are being accurate, I agree with snowyh et al...

Fig 1's area is a little short, and Fig 2's area is a little over what they should be--hence the concave/convex discussion. The difference is very small, which is why the illusion is so convincing!

In fact, if you look closely at the two diagrams, you can see that they aren't identical.

Good posting though, Jim. :)

Lizzie :)

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Jim - if you want to make the puzzle out of any material, you must make each component piece of the whole separately, to the exact length and height dimensions of each piece as shown in either original diagram - cutting up a triangle that looks or measures like the sum of the parts will fail to reveal the illusion, whereas starting with the parts to make the sums will show it, in both cases. Doing this will also reveal the mathematical solution, looking at the sums of the areas of the parts.

Ultimately, if we look at it in the form of an analogy, I think Lizzie2's description is extremely concise - it's a packaging problem!

Roz. :)

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Jim - if you want to make the puzzle out of any material, you must make each component piece of the whole separately, to the exact length and height dimensions of each piece as shown in either original diagram - cutting up a triangle that looks or measures like the sum of the parts will fail to reveal the illusion, whereas starting with the parts to make the sums will show it, in both cases. Doing this will also reveal the mathematical solution, looking at the sums of the areas of the parts.

Ultimately, if we look at it in the form of an analogy, I think Lizzie2's description is extremely concise - it's a packaging problem!

Roz. :)

Why? Why start with each individual part? Each individual part is made from the whole.

When I cut out the specific measurements, I come up with exactly the same results.

Trust me - everyone. I am not being argumentative, but rather trying to understand what you are describing. Dense is not usually a term that I am described with, so something just isn't getting through.

I built houses for a living. I can cut angles and straight lines. When I do it here, I get the same results as in the diagram. However the pieces are made, the results would be the same.

I know, I know, I'm missing something somewhere.

Thank you all for being patient with me. It is not my nature however to quit until I understand something. To spare all of the rest, my email box is open in case anyone care to expand.

I'm going to bed. Night.

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All you need to consider is the total area of the 2 L shapes in working this out. In the top diagram the area of the 2 L's is easy to work out (5x3=15 or you can count the individual squares in the background). Now look at the lower diagram and the overall outline of the shape made by the 2 L's is 8x2 this would give an area of 16 if it weren't for the missing square at the bottom. Remember the overall area has to remain the same no matter how the shapes are arranged.

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Hi Jim;

The illusion works because we assume that either whole shape can be divided up into the individual parts, and we assume that the both whole shapes are triangles with straight edges; it's only by assembling the whole shapes from precisely-specified individual parts that we can properly see the whole shapes, without making any assumptions about them.

This is an odd concept, because it challenges how to think about something - do you start with the whole, or do you start with the parts? Because we are presented with a whole shape first and foremost - albeit with rather fuzzy edges - we assume that it's a regular triangle. The only way to see thr truth about these whole shapes is to assemble them from the separately-specified parts.

As a house-builder, your skill in being able to work towards a geometrically correct whole is extremely valuable, and your ability to visualise things in this way is a true talent; I'm not a builder, but I've recently renovated my old (1925) house, which looks reasonably square to me, but there is almost nothing straight, vertical, or right-angled in it! Illusions can be very deceptive - my house looks like it has square rooms, but in fact none of the walls are even parallel! I'm sure you would see this straight away, but I can't, and it's probably because I've recently found that all is not what it seems that I am prepared to doubt the evidence of my eyes, and rely on measurment instead!

This puzzle is actually just a silly illusion, far below the advanced degrees of natural talent and finely-honed skill that you have used professionally - to paraphrase Zaphod Beeblebrox from The Hitchhikers' Guide To The Galaxy, I'm not surprised you're having trouble thinking down to this level!

...however, as a Scorpio, I know you need to know what's going on here, so do try starting with the parts - I think then you'll see that the whole shapes are what mislead us.

I look forward to hearing what you find!

Best wishes

Roz. :)

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Jim - I thought this page described the solution to the puzzle very well.

Lizzie :)

Doh!! Trust Lizzie to find the best link to the explanation :) :) :)

I did a bit more maths and found that the angles opposite the short side of each of the smaller triangles are slightly different. So it looks like the optical illusion theory is the winner after all... Time for a beer to drown my sorrows :(

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Hold the Phone!! Don't start celebrating yet.

All of this sounds good in theory, and is an easier answer. However

Since there was some question about my losing space on the saw cuts, and curved lines, I went back out to the shop, and cut out the exact pattern:

Right triangle - 5" x 13", with 90, 70 and 20 degree angles. (both the small and the large pieces have the same angles.)

This time, I used a thin piece of carboard with a sharp razor knife so that there wasn't any waste. ALL ANGLES AND LINES ARE PERFECTLY STRAIGHT.

I still get the same situation when I rearrange the pieces

There is still a one inch square left over. The grids have no bearing as long as the lines and angles are absolutely straight.

Now having meticulously cut this out to make sure that all lines are straight and angles true, I took the formula for the area of a this triangle, (13" x 5" *2 =32.5), and compared it to the total of the square inches of each individual piece, (5.5 + 12 + 7 + 7 = 31.5)

So, without the theory of curved lines and optical illusions, I still ask WHY?

And don't ask me - I still haven't figured out why, but I do believe that the answer lies in the mathematical and not the theroretical.

Oh yes - I do have pictures of this one also, but I don't think we need anymore of them. They just repeat the plywood ones.

Sorry - I really wanted the curves and illusion to be the answer. That is why I first built it myself though to check it out. Holding the cut out pieces in your own hand and arranging them gives a clearer picture. Take a piece of legal sized paper and cut it out and see for yourself.

I know you all want to shoot me by now but somehow there's an answer to this &$#@%&*%# thing. <_< <_< <_< <_< If you've had enough and want to drop out, I don't blame you. But I think the answer is still out there - somewhere.

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I'm still with the optical illusion theory (not curves though). The eye views the triangular sections of the puzzle as similar triangles ie all the angles are the same and the sides are proprtional. This is incorrect and can be calculated by working out the ratio of the horizontal to vertical sides. The red triangle has a ratio of just under 2.7:1, the green 2.5:1 and the overall shape is 2.6:1. Therefore none of the smaller triangles are similar triangles so it follows that the overall shape is not a true triangle. In fact it is not a triangle at all but a quadrilateral with 2 acute angles and one very obtuse one and 1 right angle :lol: :lol: :lol:

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Hi Jim;

Muz is absolutely correct - the angles in the triangles are not what they appear!

Using elementary geometry, where the tangent of an angle = opposite dimension / adjacent dimension:

The smallest angle in the smallest (dark green) triangle is the inverse of the tangent of 2/5, which equals 21.801 degrees;

The same angle in the larger (red) triangle is the inverse of the tangent of 3/8, which equals 20.556 degrees.

This proves that if the triangles are made according to the dimensions, the dark green triangle is "steeper", and the red one is "shallower"; thus, combined as they are in the puzzle, you will end up with the top line of the whole shape changing gradient depending on how you position the triangles - green at botton left and red at top right will give a top line to the assembled pieces that "bulges" upwards, whereas the other way round will give a top line than "dips" downwards.

Try making the pieces according to the dimensions alone, and you'll see the illusion - but the difference is so slight, that great accuracy will be required!

Roz. :)

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I'm still with the optical illusion theory (not curves though). The eye views the triangular sections of the puzzle as similar triangles ie all the angles are the same and the sides are proprtional. This is incorrect and can be calculated by working out the ratio of the horizontal to vertical sides. The red triangle has a ratio of just under 2.7:1, the green 2.5:1 and the overall shape is 2.6:1. Therefore none of the smaller triangles are similar triangles so it follows that the overall shape is not a true triangle. In fact it is not a triangle at all but a quadrilateral with 2 acute angles and one very obtuse one and 1 right angle  :lol:  :lol:  :lol:

I KNEW IT ALL ALONG

Sure I did???? Huh?

I do believe that you have it Muz. This one makes sense to me. I think others were trying to say the same thing, but you put it in language that I understood. Thank you.

I was hung up on there being a curve, so I made one myself, knowing the lines were straight. What I did not consider was the proportionality. I was looking at both triangles the same. Ha!! Excuses - Excuses!!

Darn - first time I've ever been wrong. You believe that don't you?

Thanks to all of you for having some fun with me. I'll lay off the puzzles for awhile - well, at least until a good one comes along.

-----------------------------------------------------------------------------------------------

P.S. Thanks Roz, I just saw your last post. Now it makes sense. I may not be dense, but sometimes my skull can seem rather thick.

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Hi Jim;

Sometimes it really is hard to see the wood for the trees - even for lumberjacks, I guess!

I admire your perseverance in seeking a solution!

Roz. :)

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I was hung up on there being a curve

Yeah I must admit, that bit had me thrown as well. I could see the Vertical and horizontal dimensions hadn't changed and just assumed the that the triangles must be similar because of that. ... and as for Roz's statement of elementary geometry Hah! I had to dig deep into the cobwebs of my grey matter to remember any of that. The last time I had to apply any math to anything was when I passed my professional exam for Petty Officer in the Navy, 14 years ago :P :lol:

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...erm, well, not much, O, insightful (Packaging Solution) one!

OK! I'll go and do something useful (for a change!).

Roz. :)

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. ... and as for Roz's statement of elementary geometry Hah! I had to dig deep into the cobwebs of my grey matter to remember any of that. The last time I had to apply any math to anything was when I passed my professional exam for Petty Officer in the Navy, 14 years ago  :P  :lol:

Hey Muz - Talk about cobwebs!!

I was Fire Direction Control NCO, for an 8" howitzer Field Artillery Unit in the U.S. Army, and also gave classes in firing formulas, which definitely included lines and angles.

That was 1955-1958. Then among other things, I was a licensed General Building Contractor in California for about 20 years. Oh yes, in the '60's, I also worked as a sales rep for Underwood/Olivetti typewriter and calculators.

You would think that I knew some math - Ha, you would think. I guess it is in there somewhere among all the clutter, but I just can't always find it when I need it. Darn!!

Like the old saying: "If you don't use it, you lose it." Boy, is that ever true.

More excuses - excuses.

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