Can you solve the Alice in Wonderland riddle? - Alex Gendler
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
wrong— before she loses it?
Pause the video to figure it out yourself.
Answer in 3.
Answer in 2
Answer in 1
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 Carroll—
author of Alice in Wonderland
and an accomplished mathematician
who studied this very puzzle—
one can’t believe impossible things.
