We can figure this out by finding the area of the pizza slices and seeing which is a better bang for your buck.
"How do we find the area of a part of a circle" you ask?
"Read on my dear friend!"
The area of the whole pi is π(6inches)^2= 36π. To find the area of the slice, we can set up a proportion. The slice takes up 60 degrees of the whole pi (360 degrees). 60 divided by 360 then, is the ratio of the amount of pizza that is in this slice to the amount of pizza in the whole pie. if we multiply this ratio by the area of the whole pie, we get the area of our slice! (60/360)*36π=6π.
The area of the whole pi is π(7inches)^2= 49π. To find the area of the slice, we can set up a proportion. The slice takes up 45 degrees of the whole pi (360 degrees). 45 divided by 360 then, is the ratio of the amount of pizza that is in this slice to the amount of pizza in the whole pie. if we multiply this ratio by the area of the whole pie, we get the area of our slice! (45/360)*49π=6.125π.
SO the second slice IS a tad bigger than the first slice. BUT is it worth spending 20 extra cents on?
If we divide the area of each slice by its price, we get the amount of pizza we get for each dollar we spend.
6π/$1.50=4π
6.125π/$1.70=3.6π
For every dollar we spend on the first pizza slice, we get 4π squared inches of pizza.
For every dollar we spend on the second pizza slice, we get 3.6π squared inches of pizza.
McDonald's sells chicken nuggets in packs of 4, 6, and 9 (I'm excluding 20 because mathematically, it's irrelevant here).
If the employees can only sell chicken nuggets in these amounts, what is the LARGEST number of chicken nuggets that you can order that they WON'T be able to serve you (These numbers are called Frobenius Numbers).
Well... The only way I can think to solve this problem is by seeing if it's possible or not
You can't buy 1, 2, or 3 nuggets because that won't fill a box and it says you have to sell the full box.
You can sell 4 (that's one box of 4).
You can't sell 5.
You can 6 (Thats one box of 6).
You can't sell 7.
You can sell 8 (That's two boxes of 4).
You can sell 9 (one box of 9).
You can sell 10 (one box of 6 and one box of 4)
Yan can't sell 11.
You can sell 12 (three of 4 or two of six)
You can sell 13 (one of 9 and one of 4)
You can sell 14 (one of 6 and two of 4)
You can sell 15 (one of 9 and one of 6)
You can sell 16 (two of 6 and one of 4)
...
How do you know when you're done?!?!
To be honest... I don't know how to solve it. I just realized that this math is way too complicated for me.
By the way - for the record, I'm not great at math - I just like it.
"Will I give up?" you ask.
So now, if you'd like to stick around, I'm going to jot down my internal struggle to solving this problem :)
I found this handy dandy little formula that will give you the largest amount of chicken nuggets if you only have two box amounts (once you start talking about three or more it gets complicated. maybe someday I'll understand it).
Largest Frobenius Number=(box size)( other box size)-box size-other box size
And I realized... I may not be able to solve the problem using the complicated method, but I can figure out how if a number of nuggets is possible to sell or not. Which brings me back to the question How do I know when I'm done?
If you can only only sell two of these sizes, you won't be able to sell certain amounts of chicken nuggets. BUT if you put in a third size, you're guaranteed to not have less options of amounts to sell. So... the smallest number of the three largest Frobenius Numbers should be the last number of chicken nugget requests that we'd need to test in order to conclude that we figured out Frobenius Number of Chicken Nuggets.
Here are the largest Frobenius Numbers:
(9)(6)-9-6=39
(9)(4)-9-4=23
(6)(4)-6-4=14
So theoretically, if I can find a way to combine 9 6 and 4 to be able to sell 14 chicken nuggets or less, I can figure out the largest number of chicken nuggets that you can order to piss off your Mc'Donalds cashier .
Okay... but wait. 14 is not a Frobenius number for 6 and 4 because 6+4+4=14...
And 39 is not a Frobenius number for 9 and 6 because 6+6+6+6+6+9=39.
So... Apparently, Frobenius Numbers have to be calculated using sets of numbers that have a greatest common factor (gcf) of 1.
The factors of 4 are 1 2 and 4. The factors of 6 are 1 2 3 and 6. Their gcf is 2. So the formula doesn't work, which was troublesome to me, but then I realized why it doesn't work... The answer is infinitely large. If your gcf is 2 and you can only use 6 and 4 nugget boxes, you can't sell ANY odd number of chicken nuggets (an even number plus an even number is an even number).
The factors of 6 are 1 2 3 and 6. The factors of 9 are 1 3 and 9. Their gcf is 3. So the formula doesn't work, because you can't make any values that aren't a multiple of 3.
BUT the factors of 9 are 1 3 and 9. The factors of 4 are 1 2 and 4. Their gcf is 1! So the formula works!
SO LONG STORY SHORT... If we can decide if it's possible to sell the amount of chicken nuggets ranging from 1 to 23, we will know our Frobenius Number
You can't buy 1, 2, or 3 nuggets because that won't fill a box and it says you have to sell the full box.
You can sell 4 (that's one box of 4).
You can't sell 5.
You can 6 (Thats one box of 6).
You can't sell 7.
You can sell 8 (That's two boxes of 4).
You can sell 9 (one box of 9).
You can sell 10 (one box of 6 and one box of 4)
Yan can't sell 11.
You can sell 12 (two of six)
You can sell 13 (one of 9 and one of 4)
You can sell 14 (one of 6 and two of 4)
You can sell 15 (one of 9 and one of 6)
You can sell 16 (two of 6 and one of 4)
You can sell 17 (two of 4 and one of 9)
You can sell 18 (two of 9)
You can sell 19 (one of 6, one of 4 and one of 9)
You can sell 20 (five of 4)
You can sell 21 (two of 6 and one of 9)
You can sell 22 (two of 9 and one of 4)
You can sell 23 (one of 6, two of 4 and one of 9)
FINAL ANSWER: 11 nuggets
So when you go to Mc'Donalds, ask for 11 nuggets and you'll either confuse people or get a free nugget.
Why did I do that? Because I like a challenge and I don't like taking answers if I don't understand them. Math is complicated and there are so many things we don't know and there are so many algorithms that we could simplify... but it feels so flibbing good to search for an answer and find one.
I like that there are answers that we can find and proofs that we can give in math. Because, well there aren't many subjects that you can study where the answer is black and white. And, even in math, sometimes it's not black and white. But it's amazing when you can find the answer and understand all of the work that you did.
Thank you for sticking around for that entire post. It was kinda heavy.
Mr. Del Santo, my geometry teacher in high school, would present us with riddles and puzzles. This one is my favorites from the good ol' days:
A homeless man can roll one full cigarette if he uses 10 cigarette butts. How many cigarettes can he possibly make if he collects 5,500 cigarette butts.
Well 5500÷10=550
"He can make 550 cigarettes!"
"...eh, not quite"
He has 550 cigarettes now... after he smokes those... then he'll have 550 cigarette butts that he can use those to make even more cigarettes.
550÷10=55
Now he has 55 more cigarette butts.
55÷10=5 with five remaining butts
When he smokes THOSE five he can use the remainder from before and the new butts to make one last cigarette
He can make 550+55+5+1=611 cigarettes with 5,500 cigarette butts.
This puzzle was important to me because I remember feeling tricked in class when I was asked to think about it. But I was so thrown off guard when my teacher told me I got it wrong because it was so simple. All I had to do was divide. It really made me realize that I look at problems without actually thinking about the point. What's the point of this problem? If you're homeless, and you can make use of these cigarette butts, you sure ain't gonna waste a single one. You will take every possible opportunity that you can to make the most out of nothing. This problem teaches you to think outside the box and make the most out of nothing.
Math is just as much about solving them as it is coming up with problems.
You always learn about the rules of math and how to solve problems... but every rule you'll ever learn was discovered or invented to answer a question. And my favorite part about math is coming up with questions and reading unique questions that get me really thinking.
Behind every math problem, there's a lesson. Here is a math puzzle that is fun and teaches something about what mathematics actually is.
Well... It's not 8+8+8+8+8+8+8+8
SO WHAT IT IS?!
Don't get frustrated.
Answer: 888+88+8+8+8
This puzzle is cool because it comes off as more of a riddle than a math problem... But it helps me to remember that humans didn't start out knowing everything about math. We had to figure it out and use tools that we find in clever ways to make more complicated problems work. I mean, {888...8+... +888+88+8 doesn't seem entirely unlike the systems we do know...
8+8+8+8+8+8+8+8 is the same as 8*8
and 8*8*8*8*8*8*8*8 is the same as 8^8
and 8*7*6*5*4*3*2*1 is the same as 8!
Notation is weird. Don't overlook something and say that it isn't math just because you haven't learned it or just because someone hasn't given it some form of notation yet.
This is just the beginning of a seriously long, but hopefully contained, scheduled rant about art and math. MATH IS BEAUTIFUL I TELL YOU! AND SOMETIMES ART IS VERY STRUCTURED!
I personally, think that math is discovered. And art doesn't exist without context (nothing does). So many art pieces have math in them, even if the artists didn't intend for there to be math. And there's art in math! My favorite part about math is the artistic proofs. This is just an introduction to this concept, hold tight for more math in art and art in math.
This is a super funny video about a talented man who changed his major and his life plan countless times, but always managed to create art and explore cool math concepts. It's pretty long, but don't just glance over it. It's suuuper cool (feel free to skip ahead to 14:27 and catch a super awesome closing story about seeing the world).
"There's two different ways to approach art, and they're both true. And one of them is that the world is a beautiful place and it's full of highlights and it's full of sparkles and it's full of dawns and dusks and mystery and beauty. And the other one is that it has a structure behind it, and is this structure behind it, is it mathematical? Does it have to do with sin waves and waves? And I don't know, Some does. To me, the difference between the two ways of sort of looking at the world is sort of the difference between if you are reaching out to touch the world, or whether you're letting the world touch you."
With Black Friday and Cyber Monday on the horizon, I am looking for a great nerdy gift to get my boyfriend, but you wanna know a secret?!?!? THEY ARE ALMOST IMPOSSIBLE TO FIND! You can find a lot of sites for comicon-esque merchandise and STEM kids presents (because apparently kids like science and adults don't)
For those of you who don't know, STEM stands for Science Technology Engineering and Mathematics. STEM <3
I'm looking for something...
STEM related
That I can wrap (as much as I love experience gifts, I want someTHING this year)
Useful
NOT childish
NOT a waste of desk space
Okay! So here are my finds:
Geek Apparel ALWAYS
For STEM people
S-EVERY science person has laughed about this while reading their textbook-I guarantee it.
Everyday Life Stuff
Why buy a gift that they won't ever use? Sure it's cool, but it's impractical. Here is the stuff that they will actually use that you don't even think about needing.
These Planet Bandaids are pretty cool.
Also, I don't think these actually exist yet, but ever since Biology in high school I've wanted to make mitosis bandaids. I'd be MORE than happy to see someone find these somewhere.
Subtly STEM Decór
"Can we keep it?" You drew a line when you told your roomate that you wouldn't keep rug in the shape of π in your living room... so let's get subtle. What decór is there for us nerds that you non-nerds wouldn't mind around?
Geekspeak How Life+Mathematics=Happiness by Dr. Tattersall
The Millennium Problems The Seven Greatest Unsolved Mathematical Puzzles of Our Time by Keith Devlin
Cosmos by Carl Sagan (also a super cool series on Netflix)
Why Does E=mc^2? by Brian Cox and Jeff Foreshaw
Serious Scientific Answers to Absurd Hypothetical Questions by Randall Munroe
Cooking for Geeks by Jeff Potter
F in Exams The Very Best Totally Wrong Test Answers by Richard Benson
In the past, I've gotten him...
A microscope (wayyyy on sale).
A 100% cotton lab coat (quick tip: If you're buying a lab coat for a friend who actually will use it in lab, make sure you get 100% cotton because otherwise, you could be the reason they catch fire. Trust me on this one).
Plants - cuz botany. I dunno.
I donated to The World Wildlife Foundation to "adopt an otter". They sent a certificate and a stuffed animal which was really cute because we like otters. I added these animal ones in case your STEM person is super into biology.
A pearlscale goldfish because he likes that they grow up to look like golf balls. He named her Milena (weird, I know)
Also, just throwing these in here cuz I saw them and neeeed themmm, but I think they're more girly (if you're looking for girly, etsy has really great supplies, jewelry, mugs, and stuff that is really cool):
The Fibonacci Sequence can be expressed as Fn=Fn-1 + Fn-2 . The first two numbers of this sequence are 0 and 1 or 1 and 1, but it’s sorta your choice.
So that means the pattern would be…
0
1
0+1=1
1+1=2
2+1=3
3+2=5
5+3=8
8+5=13
…
On and on forever.
It's a neat little pattern, if you're a nerd. You could've come up with it if you were really bored at your desk. As a matter of fact, when I'm bored in class, I'll just add up Fibonacci numbers and fill up pages with it. The Fibonacci Sequence is cool because as the numbers get bigger and bigger, it approaches infinity (duh), but it's approaching it at a certain rate that casues a visual phenomenon. The Fibonacci Spiral shows the rate of growth of Fibonacci numbers. The Fibonacci sprial is shown below.
If you draw the squares of the Fibonacci numbers and connect the points, like so, you create an infinite spiral. The Fibonacci Sequence is cool because of the Fibonacci Spiral. But what's cool about it is that it approaches infinity at a rate that reoccurs in nature. Most people are familiar with the Fibonacci Sequence because it occurs in nature, and there are so many more examples of math occuring in nature (which we'll talk about more later).
Here's some Fibonacci in the world.
The shape of the human face can be defined using the Fibonacci Spiral. Trees branch off in Fibonacci numbers. The number of petals on a flower are usually found in Fibonacci numbers. Plants grow with Fibonacci patterns displayed in them. Galaxies spiral in Fibonacci spirals. Seashells are seen in Fibonacci Spirals. IT'S SO FLIBBING COOL!
These two plants below have the Fibonacci Spiral going around in both directions.
