Thinking In Math

the weekend challenge! How would you solve this problem? Please share your ideas or answers in the comments.

2 years ago | [YT] | 7

Thinking In Math

Prove sketch of the shortest path problem as discussed in this video

https://youtu.be/krJf-g1Bx-s
The Shortest Distance Between A and B, with Two Reflections

Note: There are three images, representing three steps in the proof.

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2 years ago (edited) | [YT] | 3

Thinking In Math

I'd like to share my answer to one of comments on number theory books:
(original video https://www.youtube.com/watch?v=fT-zR...)

For those of you who are interested in studying elementary number theory, there are an abundance of resources available to cater to varying levels of mathematical backgrounds.

For casual learners or for a general introduction, many online materials and college-level discrete math courses for non-math majors can provide a good starting point.

For more systematic study, I'd recommend the following books, categorized according to level and focus:

For high school students or engineering-major college students seeking an introduction to the subject:

1. "Elementary Number Theory and its Applications" by Kenneth Rosen
2. "A Friendly Introduction to Number Theory" by Joseph H. Silverman
3. "Number Theory for Beginners" by Andre Weil. This is a fundamental book suitable for those with a good high school mathematics background.

For college students with a math major or a focus on math:

1. "An Introduction to the Theory of Numbers" by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery
2. "An Introduction to the Theory of Numbers" by G.H. Hardy

For readers with a background in abstract algebra or for math majors:

1. "GTM 84: A Classical Introduction to Modern Number Theory" by Kenneth Ireland and Michael Rosen
2. "Problems in Algebraic Number Theory" by M. Ram Murty and Jody Esmonde. This book presents some challenging problems in number theory.

For readers interested in applications of number theory or in computational aspects:

1. "Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity" by Manfred Schroeder
2. "Prime Numbers: A Computational Perspective" by Richard Crandall and Carl Pomerance. This book discusses many important algorithms in computational number theory.

As different books cater to different learning styles and goals, it's useful to explore a few and choose one that suits your study needs the best. Happy studying!

2 years ago (edited) | [YT] | 8

Thinking In Math

The weekend challenge. Hint: CS-inequality for fractions. see video https://youtu.be/J_sw5WnhdNc

Post your proof in the comments!

2 years ago | [YT] | 13

Thinking In Math

the weekend challenge - please provide your proof in the comments, or tune in for the video solution next week.

2 years ago | [YT] | 6

Thinking In Math

Prisoners' Probability Puzzle: Can You Help Them Gain Freedom?

Two prisoners are trapped on an island and offered a chance to gain freedom through a probability game. Each prisoner is given a fair coin. Without prior collusion, each prisoner must independently decide whether or not to use the coin. The guard will collect the coins and flip them as follows:

1. If neither prisoner uses the coin, they will remain in prison.
2. If only one prisoner uses the coin, the coin will be flipped once. If it lands heads up, both prisoners will be freed.
3. If both prisoners use the coin, both coins will be flipped. If both land heads up, both prisoners will be freed.

The prisoners are intelligent and desire freedom. They do not have the opportunity to communicate or collaborate. What is the optimal strategy for each prisoner to maximize their chance of winning the game and gaining freedom?

2 years ago | [YT] | 2

Thinking In Math

The most difficult problem in the current Number 2023 Challenges video series.

Please check out the videos here: youtube.com/playlist?list=PLX...

Problem for the weekend: Find the perfect++ number!!

A base 10 number x whose digits start with 2023. It is a perfect square. x = c^2

First part of the digits forms a perfect square a^2 for some integer a
Second part of the digits forms another perfect square b^2 for some integer b.

Find x.

x is not multiple of 10. (to avoid the trivial solutions with b = 0)
Hint: there are more than 1 solutions. You just need to find one such number.

Post your answers in the comments.

2 years ago (edited) | [YT] | 13

Thinking In Math

Please check out the videos here: youtube.com/playlist?list=PLX...

In the meantime, please send your math problem suggestions!

2 years ago | [YT] | 11

Thinking In Math

Is chatGPT correct?

Prove that product of n consecutive numbers is divisible by n!.

It tries to prove this by induction. Is the proof correct?

3 years ago (edited) | [YT] | 3

Thinking In Math

What is the equation for the curve by rotating y=x^2 by 30 degree counter clockwise?

See the red curve in the graph?

3 years ago | [YT] | 2