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Transcript for FiveThirtyEight: The Riddler solutions for September 22, 2017

Last week's riddler presented due to problems called area mazes and these were problems taken from a forthcoming book called the original area a nuisance and in a problem like this we'll give you a shape. With some of its dimensions and this is two inches of us 44 square inches. And the goal and your job was to find the missing piece in this case the height of this bottom rectangle. So this is going to be a sort of cascade of what we know and we'll move down from the top and the bottom so we know this is pointed foursquare and his and that's for instance call so we know four times something of 24 for the top this six census. We know this for singles three inches longer than the one above it so we know it six with three or non. And Reno that the threat thing on the bottom is two inches longer than that of what we ago. The bottom rectangle is 44 entrances and its eleven inches wide. So length times width. Eleven time something is 44 discussed before. And that's that this problem was very simple as giving us warmed up for the of the classic in this area masons can get pretty complicated as well. So in Ridley riddler classic things are a little more involved and same kind of idea we give you some dimensions of this funny looking fit not necessarily drawn to scale. And you go with to find. The area this box in the low. So let's start off by drawing. An imaginary line and imagining that this rectangle is filled out like and let's calculated. The entire area of this new shape so it's eleven call. And poured into slide eleven by fourteen. It is a 154. Don't went through it we can notice once we've calculated that entire area is that exactly half of its area is contained. And that's where things go what's this half of 154. The 77. And 32 point four. So half of our area it's contained. And if you shaded boxes. So one thing that I'm gonna try to prove to you is that the half of the area of this entire state is contained in the two diagonal office. And I and there. It's vertical warrant. Or this horizontal lines have to perfectly by assets. The area of the empire state let's take a different shape separate from this for the sake of this proof. And let's assume that each one of these boxes that we've divided and then to have exactly the same area. So elect before half of the area is contained in the total of those two shaded boxes. I'm not trying to prove this to you by contradiction. I'm gonna assume that what I said it isn't true and then we'll see that contradictions. So let's say we have. The extension. Now let's shift these boundaries so it's he's done we've started here let's have one vote less than one tickets and boundaries. So now we've shifted the boundaries so that neither the horizontal nor the vertical lines divide the area of this it perfectly and so I've labeled some of its new slices and let's see how it affects the area that the shaded regions defied. And so another stated reasons equal half of a total. Plus the area that I and number two which we've gains. Minus there area number one which we've lost plus the area for which regained. Minus the area history which we've lost but because of how we shifted this line we know that too is bigger than one. So we know that this term is a positive number. Similarly we games reason for. And we lost re entry but because of how we shifted this line we know for is bigger than three. We know that the positive numbers for once we've shifted. The vertical horizontal dividing lines the shaded area becomes half the total. But the positive number plus the positive number. So it's something bigger than half the total so this is our contradiction. But in this case. Our two shaded areas are half the total which means that we know are there of this. Line or this line divide the rectangle perfectly. So which one is it. Pulling over 34. Plus something here. We know that's bigger than 32 that's obvious. So we know that our vertical dividing line isn't the one that divides the shape perfectly and half. So we know has to be. This horizontal line. And we know therefore you. And Emerson area 33 squares. And those failures listens for the area mazes for last week's group.

This transcript has been automatically generated and may not be 100% accurate.