[MUSIC] Welcome back. We have the Cassini's identity here. fn plus 1 times fn minus 1 minus fn squared is equal to plus or minus one, depending on whether N is even or odd. This identity is the basis of a really interesting mathematical trick that has the name The Fibonacci Bamboozlement. I think the easiest way to show you how the Fibonacci bamboozlement works is to go to a beautiful website built by Ken Wessen. That everybody can access and I'll show you that on the video, so here's his website. So what it lets us do is, to show you the Fibonacci bamboozlement. We start with a square, so this is 8 by 8 so there's 8 rows and 8 columns of these little white squares here. And we take this square, which has an area of 64 if each area is a unit of one. And we break it up into two trapezoids and two triangles, okay? Eight is a Fibonacci number Right? And we're breaking it up to a trapezoid with the base of one, two, three, four, five. Five is a Fibonacci number and this triangle has a height of three, three is also a Fibonacci number. Okay, so this mathematical bamboozlement really has to do with the Fibonacci numbers and has to do with, in fact, Cassini's Identity. So let's see, so we have this 8 by 8 square and we move this bottom piece into this rectangle down here. We move this trapezoid into this rectangle on the top. We take this triangle and we slip it in right here and we take this triangle and we slip it in right here, okay? So, we've taken this square and rearranged it into a rectangle. And look, this square is 8x8, so this square has area 64, right, 8 x 8. For this rectangle is 5 x 13 and this rectangle then has area 65. So, somehow we've created one unit of area moving from the square into the rectangle, right? We've created one unit of the area. Do you think that's possible? Let me show you one more time, right? This one comes in here. This trapezoid goes up here. This triangle slips right in here perfectly, right and this triangle slips right in here, 64 to 65. You can become a very wealthy man if you can make things of 64 quantity into things of 65 quantity, right? So, what is going on here? This is Cassini's identity, so this is the difference Eight squared, right? Well, let's say 5 times 13, 65 minus 8 squared is equal to 1, right? This is the Cassini's Identity, so how did we manage to fit an area of 64 into an area of 65? Well we cheated or Ken Weston cheated. We can see how we cheated by clicking this honest button down here and then, we see this white area here. We can zoom in on it. This is the missing area. All along this diagonal is the missing area. We went from 64 to 65. The missing area is spread across the diagonal, which is why you couldn't really see it with your naked eye. Which makes this such a fascinating trick. Okay, this was supposed to be a favorite of Lewis Carroll who wrote Alice in Wonderland. He was an amateur mathematician also and he really enjoyed this puzzle, so what is going on here? Is the slope here? The slope of the trapezoid is different than the slope of the triangle. Right, so this area opens up as we move down. And then, when the two triangles are together, the area will stay the same, because the two triangles have the same slope, right. And then, when we move back into the trapezoid and the triangle. This missing area comes back together again, right? It turns out that these slopes are just ratios of Fibonacci numbers. They're not quite consecutive Fibonacci numbers, but there's one in the middle, right. And they're two different ratios, so when the numbers become large, the slopes converge to the same slope. So when you have more and more squares, more and more of these little squares in the big square. The bamboozlement will become better and better, okay. So, this is a rather neat puzzle. You can make it out of a piece of paper, a gridded paper and cut it up and you can see it physically. Very interesting to show children, for instance. If you happen to be a elementary school teacher, I'm sure they would enjoy it very much, okay? See you next time.
