Need to find a source for Difficult Math Problems to work for practice (Calc-2 level and above)

[FriedFish]

I've been getting bored of the problems I'm being assigned in math and I've been reading that Richard Feynman worked on a lot of creative math puzzles when he was in high school. I can't find a good repository of any math puzzles that require more imagination or thought than just solving equations tediously, so if anyone could help me find some that would be great. Preferably from a source that is free, but I'd be willing to pay a small amount of money for a book or service or something. My current math level is Calculus 2, but I'm happy with all puzzles.

 

[berkeman]

Welcome to PF.

How about looking through the hit list from a Google search for Math Olympiad problems?

https://www.google.com/search?client=firefox-b-1-d&q=practice+problems+for+math+olympiad

 

[jedishrfu]

Or the Putnam competitions for college.

https://www.maa.org/math-competitions/putnam-competition

 

[FriedFish]

These problems are interesting. My favorite kinds are the ones that seem really complex but with a subtle shift in perspective are simple, like the ones 3Blue1Brown talks about on his youtube channel.

 

[FriedFish]

Welcome to PF.

How about looking through the hit list from a Google search for Math Olympiad problems?

https://www.google.com/search?client=firefox-b-1-d&q=practice+problems+for+math+olympiad

I've been looking at these problems, and I find that there are many that I have absolutely no idea how to do. I can't tell if this is because I haven't reached a high enough level of math or if I'm just not smart enough.

 

[WWGD]

Im kind of confused. The OP joined today and was canceled today too? Don't mean to dig into anyone's private business; just wondered if there's a mistake somewhere?

 

[berkeman]

Im kind of confused. The OP joined today and was canceled today too? Don't mean to dig into anyone's private business; just wondered if there's a mistake somewhere?

This OP apparently had "forgotten" that they already had an account here at PF. The duplicate account is gone now, so they will keep posting under their already-existing account. @Interdimensional

 

[WWGD]

This OP apparently had "forgotten" that they already had an account here at PF. The duplicate account is gone now, so they will keep posting under their already-existing account. @Interdimensional

Will they post in this, our, dimension? ;).

 

[Interdimensional]

Wherever you go, there you are.

 

[WWGD]

how about just looking through an AP Calc book? Edit: @Interdimensional n
Or https://prepinsta.com/hackerrank/aptitude/algebra-and-calculus/
Or you may tweak the questions to make them harder. Ask: Where is [condition x} necessary?, etc.

 

[Interdimensional]

Didn't Feynman compete in New York City Math competitions? I wonder if I can find some of those types of problems.

 

[PeroK]

What about our own maths problems?

https://www.physicsforums.com/forums/math-proof-training-and-practice.296/

 

[Interdimensional]

What about our own maths problems?

https://www.physicsforums.com/forums/math-proof-training-and-practice.296/

Wow I never even saw this. This is great. I think they're a bit too advanced for me though at the moment.

 

[berkeman]

Wow I never even saw this. This is great. I think they're a bit too advanced for me though at the moment.

Have you looked at the MHB Math POTW forums? There are different difficulty levels...

https://www.physicsforums.com/forums/mhb-math-problem-of-the-week.361/

 

[fresh_42]

Wow I never even saw this. This is great. I think they're a bit too advanced for me though at the moment.

... and the problems + solution manual pdf for download has 533 problems. I am absolutely certain that there are many that fit the requirements. I have chosen primarily problems that teach something, e.g. theorems, inequalities, and applications. You can download them and have a look (2.4Mb).

https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/

 

[Interdimensional]

Have you looked at the MHB Math POTW forums? There are different difficulty levels...

https://www.physicsforums.com/forums/mhb-math-problem-of-the-week.361/

I've been looking at the Secondary School and Highschool problems and I still have no idea how to do most of them. Is there something wrong with me? I've always been decently advanced in math, but recently I've felt a sort of brain fog.

 

[Office_Shredder]

I've been looking at the Secondary School and Highschool problems and I still have no idea how to do most of them. Is there something wrong with me? I've always been decently advanced in math, but recently I've felt a sort of brain fog.

I honestly think those are harder than the university level problems.

 

[jedishrfu]

Checkout mathispower4u.com there are videos covering a range of standard math from middle school to first year college calculus 1,2,3 linear algebra, differential equations and statistics.

There are 5000 video solutions for many problems. You can work through each problem as described in a video, and then watch the solution provided to see if you got it right.

 

[MidgetDwarf]

I've been getting bored of the problems I'm being assigned in math and I've been reading that Richard Feynman worked on a lot of creative math puzzles when he was in high school. I can't find a good repository of any math puzzles that require more imagination or thought than just solving equations tediously, so if anyone could help me find some that would be great. Preferably from a source that is free, but I'd be willing to pay a small amount of money for a book or service or something. My current math level is Calculus 2, but I'm happy with all puzzles.

Maybe starting off with something like the Art of Problem Solving books, and working up? I was personally never interested in mathematical contest problems, but I have seen students use these for preparation, and using previous exams. There is also the yearly contest for community college math students, which I forgot the name of.

Now, math level does not equate to math ability/understanding. One can reach calculus, and a bit above with sole memorizing, if the classes are taught in the plug and chug manner (many are). Maybe you are lacking in fundamentals, Ie., geometry/trig/algebra. So that can be a good place to review.

 

[Interdimensional]

Maybe starting off with something like the Art of Problem Solving books, and working up? I was personally never interested in mathematical contest problems, but I have seen students use these for preparation, and using previous exams. There is also the yearly contest for community college math students, which I forgot the name of.

Now, math level does not equate to math ability/understanding. One can reach calculus, and a bit above with sole memorizing, if the classes are taught in the plug and chug manner (many are). Maybe you are lacking in fundamentals, Ie., geometry/trig/algebra. So that can be a good place to review.

I have heard about the Art of Problem Solving books. I'll try to get one of them. Most of the math classes I've had aren't focused on memorization but understanding of the material.

 

[James1238765]

Prime numbers have plenty of simple-looking problems that are rather beautiful (but often very hard) to solve... For example, let $\pi (n)$ be the prime counting function up to any integer n, ie.

$$ \pi(2) = 1$$
$$ \pi(3) = 2$$
$$ \pi(4) = 2$$
$$\pi(5) = 3$$
$$\pi(6) = 3$$
$$\pi(7) = 4$$
$$\pi(8) = 4$$
$$\pi(9) = 4$$
$$\pi(10) = 4$$
$$\pi(11) = 5$$
$$...$$

1. Show that $\pi (n)$ is unbounded, ie. there are infinitely many primes.

2. Show that $\pi (n) \approx \frac{n}{\log n}$, ie. the prime number theorem.

3. Show that there are infinitely many consecutive "twin" prime numbers, ie. Hardy-Littlewood conjecture (unsolved).

4. Show that the function

$$ \zeta (s) = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + ... $$

equals zero only on a certain line in the complex numbers plane $s$, ie. Riemann hypothesis (unsolved).

 

[fresh_42]

Prime numbers have plenty of simple-looking problems that are rather beautiful (but often very hard) to solve... For example, let $\pi (n)$ be the prime counting function up to any integer n, ie.

$$ \pi(2) = 1$$
$$ \pi(3) = 2$$
$$ \pi(4) = 2$$
$$\pi(5) = 3$$
$$\pi(6) = 3$$
$$\pi(7) = 4$$
$$\pi(8) = 4$$
$$\pi(9) = 4$$
$$\pi(10) = 4$$
$$\pi(11) = 5$$
$$...$$

1. Show that $\pi (n)$ is unbounded, ie. there are infinitely many primes.

2. Show that $\pi (n) \approx \frac{n}{\log n}$, ie. the prime number theorem.

3. Show that there are infinitely many consecutive "twin" prime numbers, ie. Goldbach conjecture (unsolved).

4. Show that the function

$$ \zeta (s) = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + ... $$

equals zero only on a certain line in the complex numbers plane $s$, ie. Riemann hypothesis (unsolved).

Hardy-Littlewood, not Goldbach.

 

[Interdimensional]

What would you say is the best way to develop visualization and imagination techniques for math? I'd say I already have a pretty strong imagination but I find it can sometimes be difficult to translate it to math. It works better with physics.

 

[fresh_42]

What would you say is the best way to develop visualization and imagination techniques for math? I'd say I already have a pretty strong imagination but I find it can sometimes be difficult to translate it to math. It works better with physics.

I don't think there is such a thing (for ordinary people). Different subjects require very different imaginations! Someone with good skills in the algorithmic world of numerical analysis can be totally lost in knot theory.

 

[WWGD]

How about considering one of the ( Subject) GRE prep books? If that's easy, you can consider more advanced work. Or look up qualifying exams?

 

[Interdimensional]

How about considering one of the ( Subject) GRE prep books? If that's easy, you can consider more advanced work. Or look up qualifying exams?

Interesting. I'll look at some.

 

[WWGD]

I don't think there is such a thing (for ordinary people). Different subjects require very different imaginations! Someone with good skills in the algorithmic world of numerical analysis can be totally lost in knot theory.

It's Knot Theory, it's practice.

 

[fresh_42]

It's Knot Theory, it's practice.

I thought it was Indiana Jones and the polynomial of doom.

 

[WWGD]

I thought it was Indiana Jones and the polynomial of doom.

Sorry, missed the whole series, don't catch your ref. Is that pre-reducibility days? If the polynomial doesn't split, you must acquit.

 

[fresh_42]

Sorry, missed the whole series, don't catch your ref. Is that pre-reducibility days? If the polynomial doesn't split, you must acquit.

https://en.wikipedia.org/wiki/Jones_polynomial

 

[WWGD]

https://en.wikipedia.org/wiki/Jones_polynomial

Ah, yes, I think it dealt with some Functional Analysis topics.

 

[Interdimensional]

Any other problem sources? I like to have as many wells of problems as possible.

 

[jedishrfu]

How many problems have you worked on so far? It seems pointless to keep searching if you aren't actively working on them.

 

[Interdimensional]

How many problems have you worked on so far? It seems pointless to keep searching if you aren't actively working on them.

I've solved two Putnam problems, but I'd like to find problems that are ponderable and help foster original thinking. More puzzle focused problems.

 

[MidgetDwarf]

https://www.amazon.com/dp/0486270785/?tag=pfamazon01-20

never looked at the book, so I am unsure what the audience is, or the level of problems.

the only other option I can think of, is working through an upper division probability book for these types of problems.