The hardest problem on the hardest test

That one mf that guessed and got it right🗿🗿

A friend of mine asked me this problem and the circle problem, i solved the cirlcle problem in less than 10 seconds, am i a mathematics genius?

I didn't take the Putnam in college bc I was a D-student and constantly questioned why I was pursing math if I was "no good" at it. About 3 mins into your explanation, I realized it would be 1/8 bc of how it'd scale from the 2d to 3d. That's at least 1/120 points right there lmfao

Every now and then I come back here and always end up amazed. Love this video, I'm in med school, but I participated in math olympics through out all my highschool, love maths

Me :thinking what I eat tonight

Just cool

Permutations

The expected number will always be even (or zero, if you don’t count that as an even number).

how is this guy so smart

Aleph-Math

The explanation is great and quite helpful but there is a question stuck in my mind for a while, it is that if i was one of the students who is giving this exam and got this question and i had the explanation visualized all in my mind, how should i be expressing it a handwritten answer and what all diagrams do i need to make if required??

Math is so incredible

So is the answer 1/8 or 1/32

I thought 1/8 for the tetrahedron as soon as I had the solution in the circle, but I didn't think about the crucial step of simplifying in 2D. In fact I was too lazy to search for a solution and I assumed it was too hard for me anyway (which it probably was).

if im going on the same test now I know how to cheat and finish in 1 second

I've never even expected the answer to turn out this simple

whenever i think that something i more simple than i think it is, it ends up being that the answer is 10x more complicated. so whenever i try this strategy a classic thing that happens would be like thinking the answer is so clearly 1/2 when it turnsout its the golden ratio over pi and i end up thinking "howd that happen???" however that doesnt stop me from trying since im not smart enough to understand anything else anyway

just use a calculator lmao

I still havent understood the answer

If you just put 1/8 on the test and no other work, I wonder how much credit you would get.

Idola bogeng

JEE aspirants laughing

The universe has solved all these problems.. and this all by evolution?! say yes, and you got 0 of 10^89 points.

As a med student I ain't watching allat 🤦‍♂️🤣

How did this video know I was going to want to see the written proof? I ALWAYS want to see the written proof and you rarely show it but I was especially curious this time and you seem to have predicted this haha

I took this test. I took it and threw it right in the garbage. Now I work at a Blockbuster video and I couldn’t be happier.

To show that every positive integer can be expressed as a sum of numbers in the form
2
r
3
s
2
r
3
s
, where
r
r and
s
s are non-negative integers and no summand divides another, you can use a proof by strong induction.

Basis Step: For the smallest positive integer, 1, the representation is
2
0
3
0
2
0
3
0
, which satisfies the conditions.

Inductive Hypothesis: Assume that for all positive integers
k
k where
1
≤
k
≤
n
1≤k≤n, there exists a representation using
2
r
3
s
2
r
3
s
where no summand divides another.

Inductive Step: Consider the positive integer
n
+
1
n+1. If
n
+
1
n+1 is divisible by 2, then
n
n must be odd. By the inductive hypothesis,
n
n can be represented as a sum of distinct powers of 2 and 3, say
n
=
2
r
1
3
s
1
+
2
r
2
3
s
2
+
…
+
2
r
k
3
s
k
n=2
r
1
​

3
s
1
​

+2
r
2
​

3
s
2
​

+…+2
r
k
​

3
s
k
​

, where
r
1
>
r
2
>
…
>
r
k
r
1
​
>r
2
​
>…>r
k
​
and
s
1
>
s
2
>
…
>
s
k
s
1
​
>s
2
​
>…>s
k
​
.

Now,
n
+
1
n+1 is odd, so it can be written as
n
+
1
=
2
r
1
3
s
1
+
(
2
r
1
3
s
1
+
1
)
n+1=2
r
1
​

3
s
1
​

+(2
r
1
​

3
s
1
​

+1). The second term,
2
r
1
3
s
1
+
1
2
r
1
​

3
s
1
​

+1, is not divisible by 2 or 3, so it can be represented using only powers of 2 and 3 where no summand divides another. This completes the inductive step.

By mathematical induction, every positive integer can be represented as a sum of numbers in the form
2
r
3
s
2
r
3
s
, where
r
r and
s
s are non-negative integers and no summand divides another.

For that last extra question you asked, would it be 25% of the people are circled on average? Because each person has a 75% chance of having someone cheat off of them, so the probability of no one using them to cheat would be the remaining 25%? And that would be the average for each student?

what he doin with the triagle

Well I guess I’m stupid

Another solution (maybe - I hope I haven't made any unjustified leaps): another scenario with the same probability as the target scenario is that all 4 tetrahedron points lie on the same hemisphere. Why? Because in any such configuration, 3 of the points form a plane that separates the sphere's center and the 4th point. If you take the 4th point and mirror it to the diametrically opposite point on the sphere, you have a tetrahedron that definitely contains the center. So you can see that both scenarios: the one the test asks for, and the one I suggested, have the same probability. So the question now is: what is the probability that 4 randomly selected points all lie on a hemisphere? To answer that: arbitrarily choose hemispheres (all ways of choosing hemispheres are equally likely) and calculate the probability that all points are in one of the hemispheres. That's 1/16 + 1/16 (1/16 for each hemisphere). So the answer is 1/8!

What’s the probability Burger King will be out of onion ring dipping sauce?

This question sounds like prime real estate for Bertrand's Paradox

DUDE, i literally guessed 12.5% before you started the stuff about the 2D circle only to find out i got it right all along! idek how that happened but WOAH. someone should give me one of those tests

The one who marks the test is the only one who can save humanity

Even though this problem is so difficult, it’s crazy how your 30 second answer makes it seem so obvious!

no the hardest math problem is when you are supposed to graph your answer on a graph online but it won't let you select 1.33 as a point

Videos like this make me realize how dumb I am

"Einstein: 'What I most admire about your art, is your universality. You don't say a word, yet the world understands you!'

Chaplin: 'True. But your glory is even greater! The whole world admires you, even though they don't understand a word of what you say.'"

You get it ? 😂

Is the answer for problem asked at end is 2.719....

Hmm, and if we take "1 dimensional case", which is basicly two dots, on one line and equivalent distance from center, the answer is obviously 1/2.

So, what I'm saying, can't we say that this problem for n-dimmensonal case has answer of probability 2^(-n)? That would be so freaking cool, wouldn't it?

ig 0.25 probability for the last question

=1/((1/2)^3)

So... Its a 12.5% chance

bro im 11 why tf am i watching this

구를 반으로 자르고 지름의 끝에 점A를 고정한다. 점B는 잘려진 단면의 가장자리 원을 따라 움직인다 점 B의 평균 위치는 잘려진 단면의 원 중심기준으로 점 A에 대해 90도 회전한 위치에 있다. 점C는 점 B에서 출발하여 반구의 중심을 지나 점 B의 반대편 까지 이동할수 있고 점 C의 평균 위치는 점 B에서 출발하여 잘려진 단면 원과 수직된 경로를 따라 점 B에서부터 90도 이동한 위치다 따라서 점 ABC가 만드는 평균 구의 겉 면적은 구의 1/8 이 되므로 점 D는 점 ABC가 만드는 평균 겉면적 도형의 반대편 내부에 있어야 하므로 구의 중심을 포함할 확률은 1/8 이다

when I watched the video I cannot help but think how is this nice animation made. Then I found it was done by a python library! which means basically I can also do it now. Thanks for the video and all the sharing

LLLLLLLLL, GET R3KT ChaGPT can slove any problem now. LLLLL

who creates such problems

genia!