Some schools of thought have the multiplication and division occur simultaneously, resulting in the one on the left. This method makes sense if you consider division to be multiplying by the reciprocal, IE 6*.5*(2+1)
This is essentially calculating the volume of an origin-centered half-cylinder that's capped by the surface x3+xy2.
The result is a rational number. Feel free to try to solve it. However, it baffles me a bit how it can be, given that a circle is involved.sin(18°) is famous in that its precise value can be calculated geometrically. If somebody hasn't seen or done it before, perhaps they could give a try.
Don't worry if you can't find the mistake. I only found it because I know a little bit the story from the four-color theorem. This is essentially Kempe's flawed proof of 1879, it took mathematicians eleven years to find the mistake. It's very complicated for an April fools joke. I'll refer to the written version of the "proof" instead of the video, because it is easier. Look at Figure 2b. The counterexample follows when you flip the red-green ring to pass below (circling the b-y blocks below instead instead of the y block above). Similarly, you can flip the blue-green ring and make it pass above, with the r-y blocks inside instead of only the y block. That's possible when the green block attached to the blue block marked 'b' also touches the one marked 'r'. If the rings circle this way, it is possible that the yellow-blue chain attached to the 'y' block above touches the yellow-red chain attached to the 'y' block below. When you have a blue block from the first chain touching a red one from the second, you cannot swap colors as claimed, and the argument fails. If you cannot understand it with me writing, there's a drawing on Fig. 9 in this article. It's extremely tricky, no wonder it took so long to find the error :)
I'm not sure how to reduce the size, so have this huge picture.This channel posted some weeks ago this 'joke proof' of the 4 color theorem. It seems so simple, but I've tried it for a while and I couldn't find the mistake yet: https://www.youtube.com/watch?v=Ryoipkksktg
Regions AEFCH surround the ? region with the same colors as in the video. AB is a blue/green region surrounded by red/yellow connection HDGE. Similarly, CD is a blue/yellow region surrounded by red/green connection HBGF.
The "proof" says we can switch the blue/green region AB, then the blue/yellow region CD, in order to color the ? region blue. However, if we switch the blue/green region AB, then B becomes blue. This creates the blue/yellow connection CDBE. Then switching the blue/yellow region containing C switches E to blue, making it impossible to color ? blue.
How do you color that diagram with only four colors?
Let A, B and C be the angles of a triangle, which satisfy the relation cot(A) + cot(B) + cot(C) = sqrt(3) Prove that this triangle is equilateral. Bonus points if you can do it without calculus!
On that note, both argue that phi, ie. the golden ratio, is "the most irrational number". The latter video tangentially, and perhaps serendipitously, also gives an argument for what could be considered "the second most irrational number", which would be sqrt(2). (It's all about their continued fractions, and how fast they approximate the actual value.) Indeed, it appears that if you need a "very irrational number" (that has the properties that their continued fractions have), sqrt(2) seems like an easy and good choice.
While root 2 is also fairly irrational in that sense, it's not "the second most irrational number", nor is that concept well-defined. Consider that for any number but phi, you can obtain a more irrational number by, say, prepending a 1 to the list of denominators in its continued fraction representation.
