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This magic trick, carried out with the help of six simple cards, no doubt amazed and astonished you, and made you think "Hmmm... maybe there's money to be made here!"
It turns out that The Great Fraudini can read anyone's mind -- so long as they're thinking of a number between 1 and 63, and they're willing to play a little card game with him. So let's recall how this works.
That would completely mess things up, right? We get to that in a moment.
A method of "counting by partitions" that Patricia Baggett and Andrzej Ehrenfeucht proposed at the 2011 National Math Meetings. Here's how it works:
You make a note of whether there is a leftover sheep or not -- maybe you make a mark, like a "1" or a "0"; or maybe you tie different knots in a string (more on knots in your reading). This is all you have to do to communicate the number to the King! You don't need to know how to count, to our way of thinking.
I say "Whoever's doing this..." -- I mean, "Whoever's doing this primitive counting...." This is presumably someone who doesn't know how to count -- at least not the way we do -- but they can tell if there's a sheep left over (and write "1") or not (and write "0").
Notice that we eventually have a single sheep in a pen, and that's when we're done. We have to write a "1" for the final sheep, to indicate that there's "one leftover".
Then the answer will be written as 1, 0, 1, 1, 0. That is, from the bottom up, left to right. This is important! We have to have a consistent scheme for writing.
So how do we write
Can you rebuild the tree using these "tally sticks"? If so, you can get a job in the King's counting house. Let's rebuild some....
But rather than count out fifty pennies, I'd make a stack of 10, then four more stacks of exactly the same height (one-to-one correspondence). Five stacks of 10 makes 50. This is a similar idea....
If you have a lot of pennies, you could divide them in half, count half, and then multiply by two. But if you have one left over, you'd have to add that one.
Do it again, until you get to the point where you arrive at 1 coin. Suppose you start with 81 pennies:
The secret of Fraudini's trick is another secret of the counting numbers, 1, 2, 3, 4, ....: the binary factorization of a number.
This is the second great factorization of the counting numbers. So it turns out that the counting numbers and sums of distinct powers of two can be put into one-to-one correspondence (if we count single powers of two as "sums"...).
Here are the first few powers of 2:
\[ \begin{array}{l} 2^0 = 1 \cr 2^1 = 2 \cr 2^2 = 4 \cr 2^3 = 8 \cr 2^4 = 16 \cr 2^5 = 32 \cr 2^6 = 64 \cr 2^7 = 128 \cr 2^8 = 256 \end{array} \]
(Do you see why the Great Fraudini can only read numbers up to 63?)
So in base 10 we think of "17" as \[ 17 \textrm{ (one ten and seven ones) } = 1*10 + 7*1 = 1*10^1 + 7*10^0 \]
That last bit on the right shows 17 as a sum of "weighted" powers of 10 -- "weighted" because we have to say just how many ones we need (7 of them in this case). The weight could be any of the 10 digits we need in our system,
So if I want to pay you 17 dollars, I could give you one 10 and 7 ones.
For binary numbers the "weights" are easy: 1 or 0. Just two digits. And that's perfect for computers, because they only have two fingers:
17 appears only on the 1 card and on the 16 card. We figure out how to write a number as a sum of powers of 2, and then we write that number on each of those cards. And then it will be the only number exclusively on those cards -- and the sum will tell me the number!
So we can write \[ 17=1*2^4+0*2^3+0*2^2+0*2^1+1*2^0 \] or, better yet, $17_{10} = 10001_2$ -- 17 base 10 is equal to 10001 base 2.
So if I want to pay you 17 dollars, I could give you one 16 and one 1 dollar note. I've always hoped that a student would design binary bills for me. In a way, the Fraudini cards could be considered bills, and if I need to pay you 17 dollars, I just give you each card that has a "17" on it. That would make them supremely easy to use, right?
If you're making 30,000/year, it's suddenly
111,010,100,110,000/year.
Wow! Does that feel better? You're a trillionaire!
Vi Hart has the most amazing ways of showing us interesting mathematics. She shows us that we can count to 31 on one hand. And, if you'll use the 10 fingers of two hands, you can get all the way up to $2^{10}-1=1023$...
Some people can only count to 10 with their fingers....