Not Electrical related, but can you explain this?

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I'll admit to having bitten on this. :roll: And I agree with jumper et al. on crying foul.

Do the math. 5/13 = 0.38461538461538461538461538461538. That does not equal 2/5 (.4) or 3/8 (0.375).

If the black block was really 5wide x 2tall, and the red block was really 8 wide, then if they have the same slope the red block must be 3.2 blocks tall.
Therefore if you rearrange the the two blocks, the leftover rectangle changes from 8x2 (16) to 5x3.2 (15.6) and blocks of regular size cannot actually fill both rectangles. It is not actually 5x3, that's just what the trick wants you to think. You actually have 0.6 left over. The 0.6 is the missing block, give or take.

You can posit that the red triangle is actually 8x3 and come up with a different but similar "solution".

The ratios work as a 'trick' because the values are too close to be obvious at the level of pixelation on the screen (though you can see the drawing inaccuracies if you look at it for a while, or draw it out at large scale or in CAD). It is just some visual trickery that contains no mathematically insightful 'explanation'.
 
jumper said:
I cry foul, Mivey and others mentioned slope and were led to believe that it was just the drawing.
Click to expand...

In the computer model it's not slope. They are the same. Take the puzzle apart and put it back together in the same pattern you started with.

Step 1. Line up the hypotenuses on the two triangles exactly. Make sure they touch but don't over lap.

Step 2. Position the green L block perfectly.

Step 3. Position the brown L block perfectly using the existing blocks as borders.

Now look at the bottom. Notice that there are 8 blocks with a fraction of each block extending beyond the bottom border of the triangle on the left. That is the extra space that would be added if the slopes were varied so the blocks would fit.

Same trick, different way to do it.
 
jumper said:
I cry foul, Mivey and others mentioned slope and were led to believe that it was just the drawing.
Click to expand...
Agreed. I drew it in CAD and my suspicions were confirmed. You could also use rectangles instead of squares and get close to a straight line to start with. Either way, the re-located sections do not line up even though they look lined up from a distance.

I saw no reason to post my findings as they were the same as before and had already been dismissed. But since you cried foul, I thought I would chime in and mention it.
 
K8MHZ said:
I also noticed that the puzzle that OP linked to has a straight hypotenuse but the blocks cross it, so the puzzle is still not a triangle. Actually, if you look close, it has 6 sides.
Click to expand...

OK I 'll admit it after Mivey said some blocks where hanging over here and there , well I changed a few lines.:ashamed:
 
I stand corrupted.

In addition to the above, about the computer, the slopes ARE off but not as much as in the paper model.

The difference can't be seen until 400 % zoom, but it's there. Also note that the square with the dot on it is larger than the rest by a very small amount.

Someone put some extra thought into this vs. the original.
 
If you print out two copies of the triangle and cut one out you can then overlay on the original. You can then see where the lost piece ends up. The new triangle is smaller and can easily be seen when you install the large red piece at the furthest placement . The gap of a quarter inch is seen between the red triangle and the orange and green piece.
 
In my first puzzle we have been working I threw in the orange square with the black dot to make it look more complex it had 40 squares with it and 39 without it.I think that is right? :?

Ronald:)
 
Dennis Alwon said:
If you print out two copies of the triangle and cut one out you can then overlay on the original. You can then see where the lost piece ends up. The new triangle is smaller and can easily be seen when you install the large red piece at the furthest placement . The gap of a quarter inch is seen between the red triangle and the orange and green piece.
Click to expand...

Hi Dennis

I already had the original already drawn and cut into its colored sections. I took them and arranged them as a solid tri then rearranged them with the piece missing and then drew my top line. They did not line up exactly.

My first idea about this puzzle was the top line being different was where that space was. But I always ask my self, even if that was so how could that form the empty space? I mean its not like when we rearrange the parts
the uneven spaces just jump down there and form a square, think about it. And also think about there is 40 squares in the original puzzle either way it is worked. The empty space was not a creation it was from rearranging the sections and when we seperate the orange and light green section its like stretching the puzzle by one space just in the area of the two sections its is a empty space. I might have changed my mind again.

Have a great week :Ronald:)
 
ronaldrc said:
In my first puzzle we have been working I threw in the orange square with the black dot to make it look more complex it had 40 squares with it and 39 without it.I think that is right? :?

Ronald:)
Click to expand...

That's a combo.

1) Shapes overlap

2) Triangles aren't touching

3) Slight change in slopes

A 13 x 5 triangle has 32.5 units. The L blocks take up 16, one 'triangle' is 12 and the other is 5. That's 33.

You can make the puzzle without any overlaps, I did it with card stock. By manipulating the slopes, you can make it so the triangles allow for an extra unit in the remaining space (where the L blocks go) if the slopes are convex, and take away the space of an extra unit if they are concave.

I think it's time for another drawing. Let me get some coffee.....
 
mivey said:
Same thing. If you align the bottom right of the dark green in the new configuration, the sloped line runs into the orange and does not reach the top. Real obvious at 600-700% zoom.
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Mivey, The point in my last reply was about that very point. How can the top of that slope have any thing to do with that space? That top slope is not change other than moving it from left to right above either the short arrangement or the long arrangement of the orange and light green sections.

The blank area that seems to be created from rearranging the sections is not from slight imperfections in alignment
or thicker or thicker lines, its from the separation of the orange and light green section when we move them.

One arrangement they are stacked so they need the Dark section above them when we pull them apart they need the red section above them. And also notice the red is three sections high and the dark green is two sections high
to make it match the height of the two different arrangements of the orange and light green sections.

Mark
All I am overlapping is the narrow gray outlines. like I said above that has nothing to do with forming the extra space.This is a graphic it would have to be redrawn to make it look like a extra space. Just like your pieces
you are physically rearranging on your table there, those spaces above the sloped line are drawn on those sections
they don't just jump down there and form a square when you rearrange them.That has to make sense to you.

Ronald:)
 
A picture is worth 1000 posts

View attachment 6564

The dotted line is straight and where the real hypotenuse should be.

The degrees are approximations, but close enough for this purpose.

To make a working copy of the puzzle, make a 2 x5 triangle and a 3 x 8 triangle. Then just cut the L blocks to unit size. If you try to start with a true 13 x 8 outline you will see that there isn't enough space for all the parts. You can't physically fit a 2 x 8 rectangle inside a 13 x 8 triangle. But, if you make all the parts separately and place them together, they APPEAR to form a 13 x 8 triangle. That is not the case, but it's close enough to trick most people's eye.

For accuracy, in the real puzzle, the large triangle has a 69.44 degree angle, the small one has a 68.2 degree angle. The legs are not fudged, they are whole units.
 
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ronaldrc said:
Mark
All I am overlapping is the narrow gray outlines. like I said above that has nothing to do with forming the extra space.This is a graphic it would have to be redrawn to make it look like a extra space. Just like your pieces
you are physically rearranging on your table there, those spaces above the sloped line are drawn on those sections
they don't just jump down there and form a square when you rearrange them.That has to make sense to you.

Ronald:)
Click to expand...

As I said, and I think you agree, the puzzle can be done without overlapping. It can also be done with a little overlapping. I think the original trick was done with real shapes, perhaps made of wood. I can easily duplicate it with card stock.

The reason one arrangement has more area than the other is because they are different shaped quadrilaterals. Unlike triangles, the area of a quadrilateral cannot be calculated knowing just the lengths of the sides. You also either need to know some angles, or you have to break the quad up into triangles and squares and add them up. The fact that area changes while the perimeter does not is geometric proof that the shape is NOT a triangle.

If you take 4 one inch sides and make a square, you get 1 square inch of area. If you tilt the shape so it makes a parallelogram, as it approaches total flatness, the area decreases, but all the sides stay the same (perimeter does not change). That is what is happening here. Since it looks like a triangle, it is assumed that no matter how the pieces are laid out, the area will be the same. BUT, it's NOT a triangle, so moving the parts, which changes the angle at the top, also changes the area, and does so without changing the perimeter.
 
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