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• After many adventures in Wonderland,

• Alice has once again found herself in the court

• of the temperamental Queen of Hearts.

• She's about to pass through the garden undetected,

• when she overhears the king and queen arguing.

• It's quite simple,” says the queen. “64 is the same as 65, and that's that.”

• Without thinking, Alice interjects. “Nonsense,” she says.

• If 64 were the same as 65, then it would be 65 and not 64 at all.”

• What? How dare you!” the queen huffs.

• “I'll prove it right now, and then it's off with your head!”

• Before she can protest,

• Alice is dragged toward a field with two chessboard patterns

• an 8 by 8 square and a 5 by 13 rectangle.

• As the queen claps her hands, four odd-looking soldiers approach

• and lie down next to each other, covering the first chessboard.

• Alice sees that two of them are trapezoids with non-diagonal sides measuring 5x5x3,

• while the other two are long triangles with non-diagonal sides measuring 8x3.

• See, this is 64.”

• The queen claps her hands again.

• The card soldiers get up, rearrange themselves,

• and lie down atop the second chessboard.

• And that is 65."

• Alice gasps. She's certain the soldiers didn't change size or shape

• moving from one board to the other.

• But it's a mathematical certainty that the queen must be cheating somehow.

• Can Alice wrap her head around what's wrongbefore she loses it?

• Pause the video to figure it out yourself. Answer in 3.

• Just as things aren't looking too good for Alice, she remembers her geometry,

• and looks again at the trapezoid and triangle soldier

• lying next to each other.

• They look like they cover exactly half of the rectangle,

• their edges forming one long line running from corner to corner.

• If that's true, then the slopes of their diagonal sides

• should be the same.

• But when she calculates these slopes

• using the tried and true formula "rise over run,"

• a most curious thing happens.

• The trapezoid soldier's diagonal side goes up 2 and over 5,

• giving it a slope of two fifths, or 0.4.

• The triangle soldier's diagonal, however, goes up 3 and over 8,

• making its slope three eights, or 0.375.

• They're not the same at all!

• Before the queen's guards can stop her,

• Alice drinks a bit of her shrinking potion to go in for a closer look.

• Sure enough, there's a miniscule gap between the triangles and trapezoids,

• forming a parallelogram that stretches the entire length of the board

• and accounts for the missing square.

• There's something even more curious about these numbers:

• they're all part of the Fibonacci series,

• where each number is the sum of the two preceding ones.

• Fibonacci numbers have two properties that factor in here:

• first, squaring a Fibonacci number gives you a value

• that's one more or one less

• than the product of the Fibonacci numbers on either side of it.

• In other words, 8 squared is one less than 5 times 13,

• while 5 squared is one more than 3 times 8.

• And second, the ratio between successive Fibonacci numbers is quite similar.

• So similar, in fact, that it eventually converges on the golden ratio.

• That's what allows devious royals to construct slopes

• that look deceptively similar.

• In fact, the Queen of Hearts could cobble together an analogous conundrum

• out of any four consecutive Fibonacci numbers.

• The higher they go, the more it seems like the impossible is true.

• But in the words of Lewis Carrollauthor of Alice in Wonderland

• and an accomplished mathematician who studied this very puzzle

• one can't believe impossible things.