Obviously, some of the tough and deep math may not be fit for every person, so I decided to write a more user friendly post, in which I’ll discuss some math problems that you can bring up at a non-math related dinner or something (I’m speaking from experience – people enjoy hearing these problems and ideas). These problems and ideas will hopefully change people’s antagonistic nature towards math.
Beginner: A Cute Problem in Combinatorics/Algebra:
I know that 2^64 is a 20 digit number. Don’t ask me how I know – I just know. Now, I’ll ask you a question: Does it have a digit that repeats itself 3 times? (Hint: A fifth grader can solve this problem).
Mind Blowing Idea: There are many types of infinity, some are larger than others:
We’ve all been told that infinity is a number which is larger than every number in the world. Not true. There are many different types of infinity, as a matter of fact an infinite number of infinities (haha.. but which one). That’s for a different post – but anyways, we have 2 significant and interesting types of infinities, which are א and א0, when א0<א. What do these hebrew letters represent?
א0 – the number of natural numbers, meaning, positive whole numbers
א – the number of real numbers, meaning, natural numbers, rational fractions, and irrational numbers.
Now here’s the challenge – proove or at least think about why א0<א.
Beginner: The Monty Hall Problem (a basic probability and logic question):
I’ll copy the description from wikipedia: “Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?”
The solution here is quite tricky, so try to be objective and logical. Think probability!
Beginner: For those of you who love to drive (like me):
If you love to drive, then you may like to drive fast, or slow, but what does that even mean? What exactly is speed? For those of you who love to drive stick-shift (like me), you’ll probably know what I mean when I say that I enjoy the rush of accelerating, pressing on the clutch, switching the gear, and then slowly lifting the clutch. Now what exactly is acceleration and de-acceleration? How can we represent speed and acceleration using only time and distance?
Also, if you love to drive or love the environment (again like me), then, you probably check your fuel efficiency. What exactly is fuel efficiency anyways? And why is driving very fast and breaking suddenly not fuel efficient?
Mind Blowing idea: 16=10000=121?
So, I lied a little bit. 16 is the representation of the number sixteen according to base 10, meaning, sixteen=1*10^1+6*10^0=10+6. 10000 is sixteen’s binary representation, meaning sixteen=1*2^4+0*2^3+0*2^2+0*2^1+2*2^0=2^4. 121 is sixteen’s trinary representation, meaning, sixteen=1*3^2+2*3^1+1*3^0.
Now, here’s a little something to think about: How would all of our arithmatic rules change if we were to represent a number according to a different base?
For the sharper minds, here’s an even more interesting question: how do you move between different base representations (do not read the algorithm I gave in the translation post)?
Intermediate: Risky Business:
How do we define risk mathematically? Why are some activities or decisions considered to be riskier than others?
Mind Blowing Idea: Discretization of The Continuum TIme:
Time is continuous, and this is why we can experience every single moment. This basically means that between any two given times, t1, t2, of any given interval, meaning |t2-t1|=δ seconds, for any infinitely small delta, theres a new time, t, such that t1<t<t2. This is called the completeness axiom of real numbers.
Yet, why is it that every single time that we measure time, we use a discrete measure, such as hours, years, minutes, seconds, miliseconds? Like, we never say, “I’m 24.293847983275+pi years old,” or “I can be at the office in 15.23452 minutes,” or “I can be at the office in 0.23488503 hours,” “I updated my twitter status 2.23405849843 seconds ago.” Instead, we give natural number times, such as 27 years old, 15 minutes, 2 seconds, etc. Even scientists use miliseconds, microseconds, or nanoseconds to measure, instead of using extremely long decimal representations.
So, how should we view time, as discrete or continuous? Let’s note that discrete math, such as graph theory, set theory, and algebra, is very different from continuous math, such as calculus, analysis, geometry, and topology.
Intermediate: Prove that there’s an infinite number of primes:
This sounds quite threatenting (so weird that I’m putting this in intermediate, right?), but Euclides was able to do this, and quite frankly any sixth grader can think up a proof. (Hint: Start with, let’s falsely assume that there’s a finite number of primes).
Mind Blowing Idea: Poincare’s Conjecture (proved my Grigory Perlman):
Poincare’s Conjecture (topology) states that every three dimensional space without holes can be blown into a sphere. For example, we can blow a pyramid up into a sphere if we get rid of the edges by blowing it up. Just try to visualize it. Insane!
This was an open question for nearly 100 years, and was even a millenium problem, which is a set of open questions in mathematics, and those who can prove them will receive a 1 million dollar prize. The most famous millenium problems are the Riemann Conjecture (number theory) and P vs. NP (computability). In 2010, Russian mathematician Grigory Perlman prooved Poincare’s Conjecture, but refused to accept the prize money and withdrew himself completely from the mathematical community.
Intermediate: Boys Know Girls Who Know Boys:
In my combinatorics class (this is a combinatorics question), there are 20 boys, each one of them knows exactly 4 girls, and every girl knows exactly 5 boys. How many girls are there in my combinatorics class?
Mind Blowing Idea: Representation:
This is one of my favorites, and actually one of the first things that blew my mind away when I was introduced to the math world.
So, remember the base representations that I wrote about earlier? Well, there are many different types of representations, such as:
- Representing a complex number as the sum of a real number and an imaginary number: z=a+bi, when a,b are real
- Or more generally, representing a number as the sum of a rational number and a rational number times a root, such as, z=p+q\sqrt(d), when p,q are rational, and d is whole (not necessarily natural, and when d=-1, then, \sqrt(d)=i).
- Representing a natural number as a multiplication of prime numbers
- Representing a vector as a sum of unit vectors. When we change the unit vectors, the representation also changes, according to a base change algorithm, meaning, that in 2-dimensional space, we can write: v=a(e1)+b(e2)=c(u1)+d(u2), when {e1,e2} is a base for R2 and {u1,u2} is a base for R2, meaning every vector has a represention like the one above. The smart ones can begin to think of a base change algorithm.
- Representing a continuous function as a sum of polynomials (Taylor sums) or exponents (Fourier sums).
Advanced: Cute Number Theory Question For the Clever Ones:
Actually, this question was on my number theory homework, but it can be solved without any knowledge of number theory:
Calculate the one’s digit of the number 27^(27^(27^27)).
The solution is quite tricky, but try to look for some rule and understand what exactly is the one’s digit of a number.
Mind Blowing Idea: Search:
If we have a sorted array consisting of 10,000 values, which we cannot see, and we want to know of one of those 10,000 values is 532, what’s the most efficient way to do so?
Now if that was easy, think about what’s the most efficient way to sort 10,000 random numbers. These are well-known basic algorithms in computer sciences.
Advanced: Ramsey’s Question:
Let’s assume that you can have one of two relationships with people in the world: friends or strangers, meaning, that for any given person in the world, you’re either their friend or you don’t know them, and then you two are strangers. We want to put n people in a room, such that we have 3 people, who we’ll call i, j, k, who are all friends (meaning i and j are friends, j and k are friends, and i and k are friends) or who are all strangers. By this, I mean that they were friends or strangers before they entered the room – what happens inside the room doesn’t interest us. What’s the minimal n that is required in order to ensure this?
The super sharp people can try to think about what’s the minimal n required to satisfy having m people (not 3, rather a general m) who are all friends or strangers. If you can solve this, you should publish a paper, because there currently isn’t a solution for a general m.
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