Vsauce! Kevin here, and there’s a secret message hidden in these 144 characters -- that’s
a 12x12 square grid and every card has its own coordinates. A dozen dozen, a gross. I’m
gonna decode the message randomly using dice to choose which cards I’ll turn over, and
I guarantee it will work perfectly in only 11 choices, no matter what they are -- then
I will show you how to do this with any message of any length. Even an entire randomly-rolled
book.
Here’s the setup. The grid has X and Y axes numbered 1 through 12, so the very bottom
left card is at (1, 1) and the top right card is at (12, 12). I have two 12-sided dice.
One rolls for X, one rolls for Y. There’s an equal probability of rolling everything
from a 1 to a 12. We can’t just use two normal 6-sided dice to add up to 12, because
we’d have a probability bell curve peaking around 7 and it is literally impossible to
roll a 1 with 2 d6 dice.
Alright, here we go. Roll number one. Our game begins. Okay, I got (3, 11), alright
so I’m gonna mark that card with a big X and then I’ll remove all the cards above
it, below it, to the left, and to the right -- we’re wiping out everything that shares
an X or a Y value with each roll.
GOT IT?
Let’s continue Roll number two -- (7, 6). Alright, (7,6) right here. Same thing, mark
it with an X, wipe out the column and row. I’m gonna keep playing, and if I roll a
number that corresponds to a missing coordinate, like (5,6) which is now blank, I’ll just
re-roll until I get a hit.
LISTEN. AT THIS POINT THERE’S SOMETHING YOU NEED TO KNOW. DESPERATELY.
I’m only rolling dice so that you know I’m choosing cards randomly. But when you play
this with someone, you don’t need dice, just ask the other person to choose whatever
cards they want. They’ll keep choosing until there’s only one card left without an X,
and you’ll reveal the perfectly-decoded message. And they’ll freak out. Since they’re
the ones choosing, they can be sure it’s not rigged.
More on that later -- for now let’s 80’s action movie montage the rest of my rolls.
Alright there’s only one blank card left so that gets our final X. All of our selected
coordinates are listed here on the left. I’m gonna put all of the cards in a row so that
you can read them easily when I flip them over.
WAIT. How is this even possible? I put letters on 144 cards, I randomly choose 12 of them,
and I’m left with exactly the 12 cards I want in exactly the correct order? There’s
no way this is gonna work -- RIGHT?
W...WR...WRON…. Exclamation point. WRONG!
Because this is a variation on the Magic Square discovered by Chinese mathematicians 2,200
years ago and tweaked a lot later by Canadian postman and magician Stewart James. In a Magic
Square, adding up each row, column, and diagonal gives you the same total, called the Magic
Constant. The number of rows and columns is called the order. So this is a simple Order
3 Magic Square with a Magic Constant of 15.
That’s not too hard for us now, it’s a little like sodoku -- but it took another
800 years for Varahamihira to decipher the first 16-block Order 4 Magic Square in India.
And 1,000 years after THAT, we knew there are 880 distinctly different Magic Squares
of Order 4.
Now we have pandiagonal magic squares, multimagic squares, methods for creating magic squares
of odd orders and even orders, and they’re so complex that even Euler’s hypotheses
on constructing evenly odd ordered mutually orthogonal Graeco-Latin squares turned out
to be wrong.
But the one magic square that has fascinated for centuries, even outside the math community,
is stuck in the background of an early-16th century German engraving. Albrecht Dürer’s
“Melancolia I” depicts a winged woman who represents either melancholy and depression
or geometry, depending on who you ask. She’s surrounded by all sorts of hand tools, an
hourglass, a compass, a dog... A dog is not a hand tool. A dog is your best friend. And
way in the back corner is a very unique magic square.
Dürer’s magic square isn’t technically his -- it appears in Luca Pacioli’s “De
Viribus Quantitatis” about 15 years before the engraving. Pacioli was Leonardo da Vinci’s
teacher and best friend, and the odds that Dürer coincidentally came up with the same
one on his own are astronomical. The magic constant in Dürer’s order 4 magic square
is 34 -- but the magic comes by more than just adding rows, columns, and diagonals.
You can do each of the four quadrants to get 34. The 4 blocks in the center? 34. The outside
4 corners? 34.
Inner rectangles, 34. Diagonal rhombuses, 34. The internal 3x3 blocks, 34.
Vertical X? 34 and 34. Horizontal X? 34 and 34. Trapezoids, 34. Squares that don’t use
the corners? 34.
When you sort and stack just the right combination of numbers, really incredible patterns happen
-- and it’s the same with words. The classical example is the SATOR or ROTAS square found
in Pompeii and through Europe. The palindrome TENET in the middle of the SATOR square? That’s
where Christopher Nolan’s movie title came from. Tenet. Puzzlemakers have successfully
made magic word squares up to order 9, but a true order 10 word square is still considered
the Holy Grail of recreational linguistics.
I did a modified magic square with 12, and I said you could do it even with a book of
any length. Look, here’s the mind-blowingly simple secret -- you just need to repeat your
hidden message on as many rows as there are characters in the word, message, or passage.
Hello is [H E L L O five times stacked]. You can do Happy Birthday for someone. It'd just
be 14 rows including the space. You could randomly re-create the entire Harry Potter
series, you’d just need… about a 5,327,000 x 5,327,000 grid of the book start to finish,
with a full book on each row, assuming the series’ 1,084,170 words conform to an average
word length of 4.7 characters and there’s a space in between each one. And how many
choices would it take? If however many rows or columns is equal to n, then it’ll take
n-1 choices, every single time.
No matter how big your mystery grid is, every time you make a selection and remove all the
other cards on that selection’s X and Y planes, you eliminate possible duplicates
and isolate that choice as being the only letter that can occupy that specific position
in the column or row.
It’s weirdly simple. It’s weirdly complex. It’s elegant and amazing in a strange combination
of mathematical beauty and linguistics.
And you can grab a marker and some slips of paper and play this secret message game to
reveal something monumental -- like proposing to spend the rest of your life with your significant
other (MARRY ME?).
Or... get someone to unwittingly spell the word “BUTT”
And as always, thanks for watching.
Hey! The Curiosity Box by Vsauce is celebrating its five-year anniversary. That’s five years
of delivering brain food right to your door. And it’s time for you to join us. So to
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