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 that 64 is the same as 65. Can Alice prove the queen wrong and escape unscathed? Alex Gendler shows how.
After many adventures in Wonderland, Alice has once again found herself in the temperamental Queen of Hearts court. 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 eight by 8 square and a five 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 8×3.
“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 wrong— before she loses it?
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 precisely 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 two and over 5, giving it a slope of two-fifths, or 0.4. However, the triangle soldier’s diagonal goes up three and over 8, making its pitch 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 minuscule gap between the triangles and trapezoids, forming a parallelogram that stretches the entire length of the board and accounts for the missing square.
Something even more curious about these numbers is that 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, eight squared is one less than five times 13, while five squared is one more than three times 8. And second, the ratio between consecutive Fibonacci numbers is quite similar. So similar that it eventually converges on the golden ratio. That’s what allows devious royals to construct slopes that look deceptively similar. 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 Carroll— author of Alice in Wonderland and an accomplished mathematician who studied this very puzzle— one can’t believe impossible things.