Vsauce! Kevin here, with 45 chicken nuggets,
63 cents and 130 years of algorithmic evolution that means you wait as little time as possible
in the checkout lane at the grocery store. Almost. You’ve been grinding in the Mine all day
with your pickaxe swinging from side to side… and the only thing that you value right now
more than diamonds is a box full of sweet chicken nuggets and some delectable, tangy
dipping sauce. Your mom says that she’ll get you as many nuggets as you want, but you
to make things a little difficult for her because she canceled your Realms subscription.
So, you ask her to order the largest number of nuggets that, due to available nugget configurations,
is IMPOSSIBLE for her to get -- which means that your new best friend is German mathematician
Ferdinand Frobenius. In his late-1800’s lectures, Frobenius toyed
with a thought experiment that would change how we think about nuggets forever. The premise
is simple: what’s the largest integer that cannot be expressed as a combination of integers
with the greatest common divisor of 1? Told ya it was simple. The answer is the Frobenius
number. Let me explain. Let’s say, hypothetically, that nuggets
only come in 5’s and 9’s. Their greatest common divisor is 1. If they only come in
groups of 5’s or 9’s then there are all sorts of small nugget orders you just can’t
make by combining them… like 7, 11, or 16. You can work out a table of possible combinations,
but in 1882, English mathematician J.J. Sylvester, who was so important to developing how we
think and talk about math that he coined basic terms like “matrix” and “graph,” came
up with a simple formula to find the biggest number you can’t make from two numbers that
have a greatest common divisor of 1: (x * y) - x - y. So, let's just plug in our 5 and
our 9. (5 x 9) - 5 - 9. Equals 31. There’s our nugget Frobenius. Because look. 32 is
possible that's just (9 + 9 + 9 + 5). So is 33 (9 + 9 + 5 + 5 + 5) equals 33. So is 34
(9 + 5 + 5 + 5 + 5 + 5) equals 34. And so is every other number after the Frobenius. Having a greatest common divisor of 1 is key
here, because if the greatest common divisor is higher, like 2, there can’t be a Frobenius
number at all -- and the proof of that is in the nuggets. In the U.S., McDonald’s
only sells nuggets in boxes of 4, 6, 10 and 20. Because the greatest common divisor is
2, any odd number -- very small or very large -- presents an impossible ordering combination.
In America, you just can't order 7 OR 73,212,907 nuggets. No! But as Brady Haran of Numberphile showed in
2012, the nugget situation is more complex in the United Kingdom. Since McDonald’s
in the U.K. served nuggets in quantities of 6, 9, and 20, Brady was able to stump the
cashier with an order of 43 nuggets -- the highest possible combination of 6, 9, and
20 that the McDonald's couldn’t possibly make. Because check it out, you can make 42 nuggets
because 6 x 7 = 42. You can make 44 with 20 + 9 + 9 + 6. But no combination of 6, 9, and
20 will get you 43. However, you can make every number after 43. You can't make 37,
34, 31, 28, or 25 either but 43 is the highest number you can’t make. You can work out all the possibilities for
three values of McNuggets by hand and it doesn’t take too long. But while there is a formula
we used earlier to find the Frobenius with those two numbers, there’s no simple formula
for 3 variables, or 4, or 44, or 7,218… there’s just an algorithm that ranges from
tedious to requiring a supercomputer. But who cares about chicken nugget combinations?
Is it really that important to annoy someone at a McDonald’s half an hour outside of
Barton in the Beans, or to get your mom back for canceling Realms? No -- but it matters
to anyone who uses money. Like everyone everywhere in the world. The concepts that Frobenius and Sylvester
tackled are really about mathematical optimization: what can you do with a given set of numbers,
and how easily can you do it? That’s at the heart of how we use coins and decide on
their denominations. Under our current systems in places like the US and the EU, we’re
greedy when we make change. Literally. We use what’s called a Greedy Algorithm
to process transactions -- it’s a crude, common sense way of approaching change. Basically,
we select the biggest denomination of coins to get close to a number without going over,
then the next biggest, and then the next, until we have the amount that we need. In
the US our common coins are .01, .05, .10, and .25. A penny, a nickel, a dime and a quarter.
So, to get to $0.63 cents, we’d select 2 quarters (.50), 1 dime (.10), and 3 pennies
(.03) --that's 0.63. That seems like it has to be the best possible
way with the best possible numbers. But are they really optimal denominations? In 2003, computer scientist Jeffrey Shallit
put it to the test. He found that in the 1/5/10/25 American system, the average number of coins
given as change in a transaction was 4.7. But by removing the 10 cent piece and replacing
it with an 18 cent piece, Shallit found that optimization increased markedly -- to just
3.89 coins per transaction. Knowing that removing a simple coin like the
dime surely wouldn’t be popular, he then wondered what additional denomination would
simplify transactions… and he found that adding a .32 cent piece would reduce the average
transaction down to 3.46 coins. For the Canadian system, Shallit’s addition of an .83 cent
piece would reduce the average transaction by about a coin and a half. But how easily can we even think about the
world in terms of 83’s or 18’s? And how difficult is it not to be greedy? If we had
an 18 cent coin, the best way to make .54 change would be 3 18s and not our go-to greedy
mindset of 2 quarters and 4 pennies. By employing Shallit’s optimal denominations, we’d
cut the number of coins in that transaction by half -- but how natural would it be? Is it possible that being inefficient mathematically
can be more efficient in real life? YES. If you want to get wacky with me, it’s possible
to ensure that every single change transaction between 1 and 100 cents uses no more than
2 coins. Seriously. You’d just need denominations of 1, 3, 4, 9, 11, 16, 20, 25, 30, 34, 39,
41, 46, 47, 49, and 50 -- with those, you can make change for anything with some combination
of just 2 coins. But, instead of dealing with 4 common types of coins in the US, you’d
be dealing with 16 that have no obvious rhyme or reason for their denominations. We even used to have a more mathematically
optimized coin system but rolled it back for simplicity’s sake. The US used 2 cent pieces
between 1864 and 1873, and even had 3 cent pieces between 1851 and 1889. Shallit’s
calculations showed that the presence of a 2 or 3-cent piece reduces the average coins
in a transaction by .8. It turns out that having to calculate with extra denominations
is more inconvenient than carrying around some additional pennies. With fewer coin types
you're not fumbling around looking for specific coins and holding up the checkout line at
the grocery store. It turns out that sometimes what’s best
for math isn’t best for everyday life. In 1870, French military engineer Charles
Renard proposed a series of “preferred numbers” for the world to use with… well, nearly
everything. The idea was that simple systems could streamline how we think about the world,
and a rough, rounded variation of Renard’s “R3” shows up in the “1-2-5” system
of currencies in Europe and China, while the US and Canada use a modified 1-2-5. Is that heuristic, a basic set of rules to
help us process our numerical world, the mathematically optimal way to do everything? Well… no,
it isn’t. It’s pretty good, but it’s not the best possible result in terms of math.
It’s just the best possible result in terms of people. In most of my videos I like to extract the
hidden mathematical beauty in everyday life but for this one it turned out to kinda be
the opposite. Perfectly optimal algorithms don’t always play nicely with how our brains
work and how we live our lives. There are practical limits to how we can apply our advancing
mathematical knowledge to our daily lives -- and that’s okay. Because whether it's chicken nugget boxes
or pockets full of coins what’s best mathematically may not necessarily be best for the human
experience. Unless you really, really, really want 43
chicken nuggets. And as always -- thanks for watching. Curiosity Box XII is out right now but not
for very long it's almost sold out, actually. So if you want to get yours go to CuriosityBox.com.
Now, I can't show you everything that's in here because, y'know, that would be spoiler-y
and that's part of the fun. But I will show you this. What is this? What is this space
bag? Hmm... I wonder what could be inside this space bag? Let's open it up and find
out. Duh, duh, duh, duh duh! Bumbumbumbumbumbum. This is a 3D T-shirt. That's why I'm wearing
these anaglyphic glasses. Which actually come in the box. Oh, I can't show you that. Let's
not see anything else that's in the box. But you'll get your own 3D anaglyphic glasses
and this 3D shirt. It is of the Gemini Capsule and when I look at it with my amazing glasses,
oh the sights that I see! I can see things that you can't. Well, you
could if you got Curiosity Box number twelve. Like I said, this is going to be sold out
pretty quickly. I'm not just saying that. That's just actually truth. That's actually
reality. So if you want to get Curiosity Box XII, and this amazing 3D Gemini Shirt go to
CuriosityBox.com. Also, we've updated the store there. That's
just CuriosityBox.com/Store. And there are items that were in past boxes that you can
get. There are items that have never been in a box before that are only exclusively
available at the store. So go check that out. I am going to go look 3D-rific for the rest
of my life thanks to this shirt. I like this bag too. Okay. Curiositybox.com. Bye!
