The Confuzzle

Simplicity and confusion don't often go hand in hand.  Here's a confusing puzzle, or "confuzzle", that can be made in minutes.  Although it involves simple geometric principles, it is surprising and even baffling to some people.  In short, it's a quick, easy project that is tons of fun to show others.

Here's a short video showcasing the presentation and effect of the puzzle:

Step 1 What you Need

For this project, only the following household items are necessary:

1) Two different colors of posterboard (each measuring at least 8.5" x 8.5"). 
2) Scissors
3) Pencil
4) Ruler
5) Black magic marker

Comments

[micraman]

My MICRO confuzzle at 1inch squared! (1 1/4 inch with frame)
Sorry if the photo resolution isn't good. The photo was taken with a phone cam and a magnifying glass.
Thanks for the GREAT puzzle!

[greeenpro]

Hey, that's awesome! Thanks for the contribution!!

[neomancer45]

You know, the sloping of the "not triangles" is not especially relevant, except for deluding you into thinking that the large (combined) figure is a triangle, which draws your eyes away from what is really happening inside. Take a look at fizzix18's diagrams, but ignore the yellow and green triangles. Looking at the first depiction of the blue and red shapes, their vertices denote a 3x5 grid, with an area of 15 However, the second depiction's vertices take up a 2x8 grid, which has an area of 16 This is impossible, because the red and blue shapes are respectively congruent to their previous iterations, which means that they have equal areas, which forces that one white square.

[JTreehorn]

I wonder if I am the only one completely lost right after the "not triangles" part.

[fizzix18]

My idea was NOT to give the solution, so everyone could try his one cleverness, but alas; so here 's my explanation: looking at the yellow triange, the lower left corner has a tangent of 3/8. For the green triangle this value is 2/5 and for the overall triangle the tangent is 5/13. As these three values are not equal, the three hypothenusa's are not parallel, which is conceiled by the rather thick black line. ::F::

[mrmerino]

the "overall triangle" cant have a tangent; it's not a triangle. depending on which figure you look at, its either a concave or a convex quadrilateral.

[yanLvT]

no u arent lol

[fizzix18]

Here is another well known illusion, in he same category; triangle with base 13 and height 5; if you rearrange the pieces, a gap appears. how come!

[cowscankill]

I showed this to someone a long time ago, and I have the image that proves they are the same area.

[cowscankill]

You can barely make it out, but the purple figures are the same size.

[chainsawz]

wow have you got any explaination /?

[DHagen]

Great illusion, thanks. The answer, of course, is that these are not really triangles. The apparent hypotenuse is not a straight line. In the top case, the apparent-hypotenuse is slightly concave; in the bottom case, the apparent-hypotenuse is convex. (The smallest angle in the yellow triangle is 20.56 degrees, while the smallest angle in the green triangle is 21.80 degrees. For a true-hypotenuse these would have to be exactly the same.)

[Richard5]

Neither hypotenuse is concave or convex-- The hypotenuse of the main triangle is drawn so that at the intersection of green, blue, orange in the first drawing there is a grid intersection.
The reality is that on a 13 x 5 triangle the hypotenuse at that intersection falls slightly above the grid intersection. The drawing is drawn wrong to mislead you.
If you carefully draw a  13 x 5 triangle on a 1/2" grid and then carefully place each figure in their first relative position, you will see the sliver of space above the green and orange triangles, That sliver of space amounts to one grid square. ( 1/2 sqin if measured in inches)

Rearrange the shapes and that extra 1/2 sqin shows up immediately.

Proof : (in inches) (area = 1/2 base x height)
13 x 5 main triangle = 32.5 sqin

                8 x 3 orange triangle = 12 sqin
                5 x 2 green triangle = 5 sqin
                red shape                 = 8 sqin
                blue shape               = 7 sqin

                total shapes           = 32 sqin

[j_a_s_p_e_r]

Sorry, but DHagen is right. The picture is drawn correctly.The shapes are not triangles. The one is slightly concave and the other slightly convex. Use a cad program and arrange 2 x 5 and 3 x 8 triangles. Below is an image proving the point. Click for the large image, see that if it were a triangle where the hypotenuse would have been (dashed line). Also note that the top and bottom containing rects are the same size, hence no cheating was done on the ilustration above

[bigbrosrule]

They're both concave. The hypotenuse is slighlty bent in on both so that makes them both concake.

[j_a_s_p_e_r]

Concave : A line from a vertex of the poligon to another vertex exits the poligon.

Convex : No line from any vertex to any other exits the poligon.

Top structure in my two pictures are concave. Bottom ones are convex. This is more clear in the exaggerated example (green | red) than in the (yellow | green) illustration that matches the gradients of the illusion.

The illusion is that our eyes have a hard time seeing the difference between 2/5 and 3/8 gradients and we think the two poligons are triangles

[j_a_s_p_e_r]

The illusion is revealed in an exaggerated example. I hope this settles the argument. The illusion is that these are NOT triangles, but a concave and convex structures that *look* like triangles.

[mrmerino]

well, no, they're triangles, but since the slopes of their hypoteneuses arent equal, the resulting "big triangle" is actualy bent.

[j_a_s_p_e_r]

When I am refering to "these" I am of course talking of the combined structure that appears to be a triangle in the resulting illusion. The pieces cut in the original puzzle are defined as triangles per the instructions, so there is no real ambiguity here as you imply.

The length of the hypotenuse does not directly relate to the problem, but rather it is the difference in angle of the right triangles that creates the illusion. I can easily provide a counter example where the hypotenuses are different and the illusion does not happen.

[mrmerino]

What? No, I mean, your example is very accurate, I meant that the smaller individual prices are triangles, the larger compound one isn't. I'm thinking that 3/8 is never equal 2/5, and as a result, the "hypoteneuse" is not a straight line.

[DHagen]

j_a_s_p_e_r, thanks for your patience and persistence in correctly explaining this cool illusion (the triangle illusion). Great illustrations, too.

[Electro311]

Yep. an easier way to think of it without degrees is the x length of the green triangle for 1 y length is 2.5. where as the yellow triangle is 2.66.

[Vermin]

Actually the answer has to do with packing efficiency. There are a number of ways to pack a set of regular or irregular objects. Some take more room, some less. Finding the minimum area/volume is useful for shipping cost effectiveness.

[spenfisher12]

it is hard to explain but i could make it do right

[Nextraker]

I just made one with sticky notes... wow Im amazed! this rocks!

[SpeedStrikerXLR]

hello

[greeenpro]

Thank you! Have fun.

[jagnat4000]

I thought I invented that word!!! lol Nice project

[greeenpro]

Thanks! I believe "confuzzle" is actually in the Miram-Webster dictionary now (defined as one being simultaneously confused and puzzled). Funny!

[LUCCHINA]

This is THE Winner!!!!

[greeenpro]

Haha...Thanks!! There are some great instructables to contend with on here.  Note to the world: Vote for this instructable.  There will be no raised taxes,  free quality healthcare for all, actual implementation and usage of alternative energy..ummm....no crime (none) and 27 other globally-enriching policies put into place immediately.  Oh yeah...delicious cookies for everyone too!  :)  

Seriously...best of luck to all of the other authors. 

[Frivolous Engineering]

My brain just melted!

[hammer9876]

Eat ice cream. Typically induces brain freeze. That should fix a melted brain. Tastes good, too!

[greeenpro]

Hahaha...I'm sure there's an instructable on here to fix that problem.

[3BricksHigher]

Dude! Great illusion. Also thanks for the props :) Very much appreciated!!

[greeenpro]

Thanks for the idea!

[cowscankill]

At first I was thinking "Sooo...? You just rotated the pieces..." The second time I looked at it, I thought "WHOA WHAT THE HECK?" Yeh, I still don't know how it can fit in the same surface area xD

[Arano]

they don't ;)

[cowscankill]

Ah yeah, after a while of staring at a triangle example, I finally got it. Haha.

[wibrle]

Wow! Thats a cool trick. :)