The Puzzle from Hell

This is interesting! Go back to the floor tile challenge! I did a load of graphical representations! ( not the posted ones but others using only the prime numbers) What we have here is a 13 by 5 tiled floor, with a diagonal line passing from one corner to the other! The diagram is very clever in rounding up the position of the acute angle of the dark green triangle to meet at a line when in effect, it doesn't!

This is the bit that is missing!

Simple.

Draw a line from the lower left corner to the upper right corner of each figure. You will see that the top figure's edge is just below that line. The bottom figure's edge is above the line where the greed and red triangles meet.

The delta area above and below the line is the empty block in the lower figure.

Now you have been to Hell and back.

Hi Yuval,

I posted this one up on CR4 some time ago.

Take a very close look at the hypotenuse of the enclosing triangles. It looks like a straight line, but it actually isn't. In the top configuration, the "hypotenuse" bends inward slightly. In the lower configuration it bends outward. The net difference in area between these two triangles is one square unit, which accounts for the gap in the lower triangle.

One way to make the bend more obvious is to change the vertical scale to (say) two units high for every unit across. It will become readily apparent that these two objects are actually four-sided polygons, not triangles.

Glad you enjoyed it!

Yes, according to the above illustrated AutoCad figure, it is possible to see (and understand given the difference in angles and fragmented hypotenuse), that the line "concaves" to the meeting point.

A. True, 4-facet polygon, not rectangle.

B. Did you get to measure this difference of area?

Compute the area of a true triangle having the same vertices as the ones shown in the picture. Then compute the net area of the pieces that make up one of the figures (either one). Do not include the area of the gap. Subtract the net area of the pieces from the area of the true triangle. It will be 1/2 unit less for the upper "triangle." When you add the area of the gap, the lower "triangle" will have 1/2 unit more area than the true triangle. To wit:

Area of true triangle: (5 x 13)/2 = 65/2 units = 32.5 units

Area of:

Red: 12 units

Cyan: 5 units

Green: 8 units

Yellow/Orange: 7 units

Sum of areas of pieces: 32 units (area of upper "triangle"). 1/2 unit less than true.

Adding the area of the gap (1 unit) to the area of the pieces in the lower "triangle" gives: 33 units. 1/2 unit greater than true.

Subtracting the net areas gives us one unit, the area of the gap. Our comparison with a true triangle shows us that the difference in area is divided evenly between the two figures.

Shit!... Another one bites the dust

A while back, Europium and I even talked about developing a free energy scam based on this puzzle. Something like: Clearly different arrangements of these magic shapes result in different amounts of matter. If the amount of matter changes, then the amount of energy must change per e=mc2. Although the differences in matter appear small (just one unit in 30 or so) we are all aware that c is a very big number, and c squared is an even bigger number...

Maybe by adding some matter you could reach C square (and be the first to shorten the space barrier) for travel.

Yeah. I know. A Zelebe Drek...

Not that some clever canvasing wouldn't entice people to invest in the research of it, because I've seen even more outrages invested into...

I guess it's all a matter of presentation

Hehe.

Years ago my mom mentioned something about wanting to re-arrange the living room furniture to make the room look more spacious. On cue and with tongue planted firmly in cheek, I sketched these two figures on quadrille paper and pointed out that certain kinds of re-arrangements can produce literally more floor space. Such arrangements not only make the available floor space more usable, but they actually create space out of nothing. "It all has to do with the unique geometry of triangles," I lied. She countered by pointing out that most rooms are rectangular. "Not to worry, mom. Rectangles can be tessellated into two triangles, leading to a doubling of the additional floor space over using this technique with one triangle alone."

I'm going to Hell for that. I'm certain of it.

I'm going to Hell for that. I'm certain of it.

I hope to meet you there... just not anytime soon.

I believe we are in Hell right now and we have all met already.......

You confirmed my suspicion!

Would you like your brimstone medium rare or well done?

Let me guess:

If hell is right here and right now on earth, then Heaven must be here and now, as well...

Well,

You confirmed mine!

I feel that it comes later, after death......that is my belief.

And I always thought that symmetry is a divine law

When I was a kid, we were really close to understanding matter and its elemental particles. There were just three, and the electrons orbited the neutrons and protons in pretty well defined ways. Later, I found out that the scale shown in my textbooks was all wrong: If the nucleus was the size of a period at the end of a sentence, then the first orbit would be somewhere in the next town over. And then I came to "know" that these electrons are not the elemental particles at all, and that the "orbits" are a little more like clouds. Clearly, what I "know" now is all wrong, even at the simplest most elemental level. How does life (zillions of times more complex than the simplest particles) work? I can only make the crudest guesses -- it's got to be magic, as far as I can tell.

Then... going further out beyond the things we try to understand by repeated observations (i.e., through science) there are the metaphysical questions, where the layers upon layers of abstraction make it impossible to say anything with more than .0001% certainty. Is Rembrandt better than Picasso? Can there possible be an answer to such a question? Is heaven here, there or elsewhere? How could one possibly answer?

For me, our choices in the here and now put us into heaven or hell on earth. That's plenty abstract enough for me. Once I get to the point that I think I understand 1% of what happens and can be observed here on earth, then I'll work on the metaphysics. At my current rate of progress, I figure by 3050, I might be getting close to starting my study of metaphysics.

In the mean time, I try to be nice, not take more than my share, to say "I love you" frequently, and to lend a hand to those in need. I take it on faith (along with a fair bit of observation) that doing so will help to keep me from creating my own hell on earth.

"...Is Rembrandt better than Picasso?..."

Or, to an alien: Is Rembrandt that much different from Picasso?

Hi Europium

I think I got to know about CR4 because of your post.

What a vivid demonstration !

Picture worth a thousand words.

Just another jigsaw puzzle , where broken pieces will fit in there respective position only , and not the way around........civil engineers ......step in.......

Nope. The solution you show is the "Elegant" one.

2:5 = 40% grade, and3:8 = 37.5% grade by inspection.

When the pieces are rearranged and a new line added from the upper to lower points of the triangle the missing unit square appears as an long sliver with 1 sq.unit area.