# Hopscotch Programming Puzzles

#63

That’s like 10^83 or something, right?

And plank volumes to fit in the universe is 10^180

Idk, there’s a Numberphile video on this somewhere..
There’s also a Numberphile video about Graham’s number
And there is also this about Graham’s number. (Sorry, this is a tiny bit of raunchiness. But the whole math is great to ponder about) (I love googology. I think I’ve told you that, though)

#64

@BuildASnowman pls go die in a hole. actually, pls go die in one of the zero’s of your percentage of the sums of something i dont care about.

@JonnyGamer good point. i have been blinded by my own ignorance. there was legit part of me that went “so… since 4+++4 doesn’t make much sense, then 4!!! will not either!” how am i gonna last through high school?

actually, i just suck at number theory and combinatorics. i can do algebra. and a lil geometry.

STORYTIME:

the story duh

im asian so mommy makes me do math. mommy says im bad at math (asian standards. ugh) so mommy sends me to smart math kid camp. (It wasn’t that bad tho. Everyone loved me, especially the teachers. except they were probably annoyed by the end cause i kept singing Low by Florida and dancing to Despacito.) anyways… mommy sends me here so i can get better at math. i say camp is for making friends. Day 1: I see a friend i know and I sit with her. (same day) the teacher separates us cause we be talking too much. whoopsies Day 2-4: screw the teacher, im still sitting with my friend.

Day 5: its time to make new friends. i see a kid named oscar, who is wicked good at math and naive about the whole world, but he is sitting alone so i sit next to him so his smartness can rub off on me. the lesson today is number theory. so on day 5, i don’t do any work and Oscar feeds me the answers which is pretty nice. So there’s this problem that the teacher makes us do, and its:

During a recent math contest, Sophy Moore made the mistake of thinking that 133 is a prime number. Fresh Mann replied, “To test whether a number is divisible by 7, we just need to check that the sum of the digits is a multiple of 7, so 133 is clearly divisible by 7.” Although his general principle is false, 133 is indeed divisible by 7. How many three-digit numbers are divisible by 7 and have three (pairwise) different digits with their sum divisible by 7?

And this Oscar kid BASHES OUT EVERY SINGLE NUMBER WHOSE DIGITS ADD UP TO 7 and divides each one of those numbers by 7 to get the answer. and there’s like 60 numbers and he does it out in like 5 minutes.
Day 6: still sitting next to oscar hoping i will be smart like him by having him feed me all of the answers to the problems. Oscar proves by hand that 1111111 (7 1’s) is prime.

Anyways i sat with three other kids, two of which scared me (one wouldn’t talk and the other kept swearing) and the third was charlie and he was so adorable and huggable.

#65

Hehe, screw the teacher That’s hilarious
That sounds like a super fun (and interesting) math club!

Also, here are some tips about the “7’s” problem! Find the first number above 100 that is divisible by 7. Then, all you have to do is add by 7 and check if the numbers are divisible by 7 as well. That’s basically how it works (he probably did that)

Lastly, 1111111 is sadly not prime, it is 239 * 4649
Actually, repunit primes are really fascinating! The only known ones have 2, 19, 23, 317, and 1031 digits. (Prime numbers are awesome)

Hehe, did you notice Sophy more is a pun on Sophomore (10th grade). Hehe, at least they are inventing good names instead of names like Mashoon in the Commor Core books. (Actually, Mashoon is a pretty sweet name )

Haha, I have a whole bunch of hilarious math class stories
One time I was showing my friends Hopscotch and all of MP’s amazing projects and the teacher came over and was like, “What is thiiiiiis???” and we were all like, “It’s a coding program” and he started playing the castle game by MP and he didn’t give my iPad back until the next day

He also spilled his coffee on his computer in the same class because of the fire alarm, that was hilarious

#66

HAHAHAA Oscar spent like thirty minutes on that 1111111 problem. I feel so bad for him.

And also, he didn’t know the divi illite rule so he legit divided every number by seven.

#67

Woah, there’s like 900 3digit numbers. That’s quite a lot

#68

i meant all three-digit numbers whose digits added up to a multiple of seven.

#69

Ah, ok, that’s still a looot though

#70

@BuildASnowman

Have you heard of inter universal geometry?

#71

Yes I have! Of course I, like other mathematicians, am nowhere close to understanding Mochizuki’s work, but I at least have an idea of the math behind it. It’s by far the most difficult math I’ve ever learned, but it’s absolutely beautiful and I would love to talk with you about it. How do you know about it? Do you have questions about it?

#72

I just recently watched a Numberphile video about the abc Conjecture, and the Conjecture was invented by the guy doing the inter universal geometry.

It seems really cool, apparently it’s like a whole new perspective on math, and also if the abc conjecture is proven, it will either prove or disprove a great many other conjectures and theorems. They were talking about how all the great mathemetitians are still going through it trying to understand what in the world (or should I say universe) he had just invented. Anyways, I know nothing about it myself, but it seems like something that would be awesome to explore

By the way, I finally figured out the term of mathematics that’s based on the fun/puzzles/inventing/conjecturing. It’s called recreational mathematics (there was a book called: A journal of recreational mathematics, which was a series of magazines that started back in 1968 and ended in 2014. I’ve been trying to find some of these magazine/booklets because it sounds absolutely incredible)

#73

I want to try to explain it to you. And it’s gonna be difficult, that’s for sure. Take a look at this paper for an idea of what an “introductory survey of the philosophy behind it” looks like. But I think I can do it, because you’re brilliant and it’s beautiful. Give me a few minutes

#74

So, let me first start out with some geometry. I will introduce the notion of a “fundamental group”. First, take any surface. Could be a plane, could be a torus, could be a sphere, whatever. The fundamental group of that surface is the set of all closed loops that can’t be deformed into eachother. By “deformed”, I mean squeezed, rescaled, moved around, pulled, whatever. Like a rubber band.

Now, to take a primitive example - a plane. Any loop you make can be deformed into any other loop, so the fundamental group has one element (which is a function that describes the loop but we won’t get into that). If you take a torus, it’s more complicated - there are two loops, the red and the blue:

You can’t stretch and squeeze the red to make the blue, and vice-versa, so the fundamental group of a torus has two elements.

Here’s where it get’s tricky - inter universal geometry isn’t studying fundamental groups. It’s studying etale fundamental groups. It seems like a small change but it isn’t - it’s a lot trickier. You know how we defined our fundamental group by “deforming” loops? Well. it turns out there is a type of “loop transformation” that’s stronger than deformation - they’re called “deck transformations”. You don’t need to know what they are, but you just need to know that if we have a fundamental group, and we take any loop from it, a deck transformation allows us to transform that loop into any other loop in the fundamental group.

What this means is that the deck transformation defines the fundamental group - all you have to do is start out with one loop and just apply every deck transformation to it and you have the rest of the fundamental group. In this way, we can define the fundamental group of some surface as the set of deck transformations of one of it’s loops (this is just restating what I said in the previous sentence, but in a cleaner way). An etale fundamental group is what happens if you replace the “deck transformation” with something else:

Etale Fundamental Group of a surface = The set of etale transformations1 of one it’s loops2

Here, an etale transformation is simply a transformation that preserves the structure of the loop, which is a fancy way of saying that the loop can change, but some of it’s fundamental properties can’t (imagine a real rubber band - you can stretch it, sure, but not past a certain point, and you can’t really shape it any way you want - try making the shape of a giraffe with one :P)

Now, we have defined the etale fundamental group of a surface. What does this have to do with the abc conjecture? Well, the abc conjecture is equivalent to another problem called Szpiro’s conjecture. Szpiros conjecture deals with elliptic curves, which are simply equations of the form y^2=x^3+ax+b. Szpiro’s conjecture establishes an inequality between two numbers having to do with elliptic curves - the discriminant, and the conductor. You don’t need to know what these two numbers are (I’ll be happy to explain if you want), nor what Szpiro’s conjecture says. You just need to know that Szpiros conjecture deals with elliptic curves, and that elliptic curves are just that - curves. Just like a plot of x=y is a line, elliptic curves make a curve:

Szpiros Conjecture names a certain property, let’s call it “Property Szpiro” (remember this, you’ll need it later), and then says that all elliptic curves have Property Szpiro. Of course, there is no proof, because it’s a conjecture. And it turns out that Szpiro’s conjecture is equivalent to the abc conjecutre.

Now, we can think of the ‘curve’ that the graph of an elliptic curve draws out as an object, and actually take the etale fundamental group of it. Now, here, I must digress, my analogies break down. You can’t think of “loops” on a 1 dimensional curve. But, you can almost think of it as a “hack”. We define the etale fundamental group of a surface, and we formalize it mathematically, and then we realize that we can plug in an elliptic curve into it and actually get something out. Just like if you write a program to calculuate the nth fibionacci number, and suddenly you realize you can plug in negative numbers. Obviously, there is no such thing as the -2nd fibionacci number, but the “negative fibionaccis” have a pattern nonetheless. In our case, the etale fundamental group was made for surfaces, but because of the way that mathematicians formalized it, we can take the etale fundamental group of an elliptic curve and it will give us something that makes sense.

Here is where everything comes together. Mochizuki’s work is simply answering the following question:
Given an etale fundamental group of some unknown curve, how much can we figure out about the curve?

In fact, even more specifically:
Given an etale fundamental group of some unknown curve, can we re-construct the entire curve?

That’s what inter-universal geometry is. Constructing elliptic curves (or more generally, any curve) from their etale fundamental group only.
.
Mochizuki’s work gives an affirmative answer to that question (if his work is correct, of course). He then uses this to prove the abc conjecture by:

1. Giving a way to construct an elliptic curve from it’s etale fundamental group.
2. Reminding us that all etale fundamental groups have a certain property, let’s call it “Property X”. This wasn’t a new property, it was something that is pretty clear from the formal definition of etale groups.
3. Morphing the set of all etale fundamental groups into the set of all elliptic curves using his method.
4. Showing that “Property X”, when put through his process, becomes “Property Szpiro” (remember that?). Therefore, all elliptic curves have Property Szpiro.
5. All elliptic curves having Property Szpiro means Szpiro’s conjecture is true, meaning the abc conjecture is true.

Of course, this is only the motivation behind inter universal geometry - how any of this is actually accomplished is a lot more difficult. If anyone is interested in learning, though, take a look a few short surveys that give you an idea of the math involved. Be warned, you need a strong background in class field theory and galois theory to understand any of it. As a stepping stone, try learning about the Neukirch-Uchida theorem and the Hodge-Arakelov theory of elliptic curves.

1 Actually, they’re called “etale morphisms”, but morphism basically means transformation, so I called it a transformation to keep the terminology consistent.

2This is kind of imprecise, just because the etale fundamental group is so hard to define. Technically, it’s the inverse limit of finite automorphism groups. But even if you know what that means, that gives you no intuition as to what the etale fundamental group is.

#75

If you have any questions, no matter how stupid, please ask them. That goes for everyone. I will be happy to answer. Even to say “I didn’t understand a thing you said” (Although I would prefer you be a bit more specific :P).

#76

Awesome! I’m going to spend time reading this, I can’t do it right now, I’m going to be on an airplane tomorrow, so I’ll probably respond anytime tomorrow (probably evening)

I really want to thoroughly read it so I can grasp the concept (it looks so exciting! )

– Also, thank you so much for spending and hour and a half on answering my question! It means a lot!

#77

No problem at all. I love these types of things - it’s sad that there aren’t resources available for people without background in Galois Theory and all types of other weird math to understand this kind of stuff. Take your time, read it through a few times. I’ll correct any wording that I find to be confusing, and ask me as many questions as you want.

#78

Sorry to be kind of unrelated, but since all of you all seem like math geniuses, do any of you know any good websites that teach math in easy ways?

I am not a math person. I don’t care about advanced stuff. I just want to get through algebra this year because I struggled last year. (I was fine, ended with a low A, but I’m not comfortable in math)
I also have difficulty working fast enough on tests and stuff. Any math test tips?

My school has no advanced track, everyone takes algebra 1 in 7th grade and continues on it for 8th.

Want more Math? (Math Resources)
#79

Purplemath has some great algebra topics.
And this may sound obvious, but if you don’t understand it, just google “(topic name) + explanation”, or “(topic name) + intuition”, that’s how I got through a lot of difficult concepts.

As for math taking tips, work on arithmetic in your head. In my experience, the difference between a fast test taker and a slow test taker is whether you work out everything on paper or you can do most of in your head. You should know the whole times table up to 12*12 and be able to do any multiplication within it in a fraction of a second. Also, unless you need a decimal in your answer, never do long division. Take a guess at what the answer is and see if it works, if you need to go lower or higher, etc.

For example, my thought process if I want 4420 divided by 13, would be - 13x100=1300 and three times that 3900, which is 13x300 and pretty close to 4420, but under. So I try 13350, which I on paper can work out to be 4550 in about 5-10 seconds. That’s just 130 over, so the answer must be 4420=13340, so 442/13=340. That whole thing takes me about 10-15 seconds to do, which is a lot shorter than doing the long division.

It’s all about little efficiency tricks like that here and there. Figure out the most intuitive way to do algebra for you, or the most intuitive way to simplify radicals. That’s how you get speed - don’t do it the hard way and get confused.

#80

Sorry to interrupt, I think your discussion is great but I think you should move to the Math on Paper Compared to iPad (Addition, subtraction, algebra, etc.) topic or The Numberphile Topic – The Official Nerdy Math Topic!, as this is Hopscotch Programming Puzzles xD

#81

Wut my post is linked cool

#82

Good idea If codingcupcake wants to respond, we can continue our discussion there.