Other Where Can I Find Challenging Math Puzzles for Calculus 2 and Beyond?

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The discussion centers on finding engaging and imaginative math puzzles beyond standard equations, inspired by Richard Feynman's creative approach to math. Participants suggest various sources, including Math Olympiad problems, Putnam competition challenges, and resources like the Art of Problem Solving books. Some express feelings of confusion and frustration regarding their current math abilities, questioning whether they are lacking foundational knowledge. Others recommend exploring different problem types, including prime number theories and visualization techniques, to enhance creativity in math. The conversation highlights the importance of finding stimulating problems to rekindle interest and foster original thinking in mathematics.
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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.
 
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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.
 
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?
 
WWGD said:
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
 
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berkeman said:
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? ;).
 
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Wherever you go, there you are.
 
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  • #11
Didn't Feynman compete in New York City Math competitions? I wonder if I can find some of those types of problems.
 
  • #15
Interdimensional said:
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/
 
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  • #16
berkeman said:
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.
 
  • #17
Interdimensional said:
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.
 
  • #18
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.
 
  • #19
FriedFish said:
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.
 
  • #20
MidgetDwarf said:
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.
 
  • #21
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).
 
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  • #22
James1238765 said:
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.
 
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  • #23
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.
 
  • #24
Interdimensional said:
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.
 
  • #25
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?
 
  • #26
WWGD said:
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.
 
  • #27
fresh_42 said:
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.
 
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  • #28
WWGD said:
It's Knot Theory, it's practice.
I thought it was Indiana Jones and the polynomial of doom.
 
  • #29
fresh_42 said:
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.
 
  • #32
Any other problem sources? I like to have as many wells of problems as possible.
 
  • #33
How many problems have you worked on so far? It seems pointless to keep searching if you aren't actively working on them.
 
  • #34
jedishrfu said:
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.
 
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  • #35
  • #36
The reason I posted this thread was because recently I have felt strangely around math. Over the years I've been called exceptionally good at math by most of my teachers, but for some reason lately I've felt slow and unimaginative. I can't quite tell if this is just my brain annoying itself, or if I lost something I had before or if I ever even had something. I've also found the problems i've been assigned pretty boring. What have you guys done in your experience with this?
 
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  • #37
Interdimensional said:
The reason I posted this thread was because recently I have felt strangely around math. Over the years I've been called exceptionally good at math by most of my teachers, but for some reason lately I've felt slow and unimaginative. I can't quite tell if this is just my brain annoying itself, or if I lost something I had before or if I ever even had something. What have you guys done in your experience with this?
Can you give an example so that we can know what you mean when you say "math"?
 
  • #38
Recently with problems that require more creative and original thinking I feel like I've been less capable than I used to be. Kind of like a brain fog.
 
  • #39
Interdimensional said:
Recently with problems that require more creative and original thinking I feel like I've been less capable than I used to be. Kind of like a brain fog.
That's not an example. You could likewise tell your age, your level of education, and your goals. Are we talking about graduated math or highschool math?
 
  • #40
What do you mean by example? A specific problem? Or an area of math? EDIT: I see your edit.
 
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  • #41
Interdimensional said:
What do you mean by example? A specific problem? Or an area of math?
Different levels of education (high school, college, undergraduate, graduate) require different skills.
Different areas in mathematics (geometry, calculus, algebra) require different skills.
Different goals (hobby, school, study, research) require different skills.
 
  • #42
fresh_42 said:
Different levels of education (high school, college, undergraduate, graduate) require different skills.
Different areas in mathematics (geometry, calculus, algebra) require different skills.
Different goals (hobby, school, study, research) require different skills.
Oh I see. So your saying that different areas of math and education and goals require different skills. That's comforting to know. I partially thought I was losing my mind.
 
  • #43
Currently I'm in Calculus 2 but in a high school class. Eventually I'd like to get into physics research.
 
  • #44
Interdimensional said:
Currently I'm in Calculus 2 but in a high school class. Eventually I'd like to get into physics research.
Could you give us an idea of what topics you've covered in Calculus 2? Just make a list.

-Dan
 
  • #45
Sure.

Calculus 1 - of course

Calculus 2 - numerical integration, which we skipped in calculus 1 and went back around to, integration of natural logs and exponential functions, trigonometric integration, and starting integration and differentiation of hyperbolic functions.
 
  • #46
My math teacher used to say: "Everybody can differentiate but it takes an artist to integrate."

Means for us ordinary people: practice, practice, practice. Learn trick after trick: Weierstraß substitution, additive symmetry, multiplicative symmetry etc.

I like this book for that purpose:
https://www.amazon.com/dp/0846407612/?tag=pfamazon01-20

... but it seems that it is currently unavailable or very expensive. However, there are similar ones. Just look for a) a Russian author and b) written between 1950 and 1970. These criteria guarantee that you find a book close to real problems with numbers and techniques e.g. integration. The soviet study of mathematics was primarily targeted to create engineers.
 
  • #47
Interesting. I have always found integration to be a more enjoyable process than differentiation. I've heard similar things about Hungarian mathematics books as well. Are there any translation difficulties I should know about?
 
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  • #48
Interdimensional said:
Interesting. I have always found integration to be a more enjoyable process than differentiation. I've heard similar things about Hungarian mathematics books as well. Are there any translation difficulties I should know about?
The book I linked to is in English. It contains example after example, so it's not a textbook to study calculus.
 
  • #49
Take a look at the Project Euler problems, too. There is a variety of difficulties. Many of the problems are computing based, but you have to figure out how to set up the math for the computer to get the correct answer.
 
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  • #50
The Art of Problem Solving Volumes 1 and 2 cover basic algebra, geometry, and precalculus, but the problems are challenging enough to take you to the contest level.
When you take Linear algebra, look at Halmos' Problem Book
For interesting integration problems, watch blackpenredpen (and of course try to work out the problems before he does. For books on integration, see Inside Interesting Integrals by Nahin, Irresistible Integrals by Boris and Moll, and (Almost) Impossible Integrals by Valean

Are you in the US, @Interdimensional ? What grade are you in?

Do you want physics problem books as well?
 
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