Explaining the bizarre pattern in making change for a googol dollars (infinite generating functions)
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it whenever someone asks about coin change counts or generating functions, since it works out an explicit formula for making change for n dollars.
Recommended by 2 people across 5 different months, 2019 to 2022.
What it is
A Mathologer video that counts the ways to make change for a googol (10^100) dollars with standard US coins and explains the strange patterns in the answer. It builds up from a visual algebra approach to infinite generating functions, with no prior knowledge assumed.
What people said
Strong rec Sayan Dutta · 2021-07 · Mathematics Stack Exchange · on “How many similar triangles with positive integer angles?”
@peawormsworth And, here's an excellent video that discusses a completely different topic, but goes through this part in the first few minutes - [this video]
Recommends awkward · 2019-03 · Mathematics Stack Exchange · on “How can I learn about generating functions?”
Here are some resources to get you started on generating functions. With one exception, which is clearly designated, any of the items mentioned here should be suitable to provide a gentle introduction to GFs for total newbies. generatingfunctionology by Herbert S. Wilf is probably the best introductory text. You can find this book in pdf format for free, online, but I think it's worth adding a hard copy to your library. The writing style is breezy and entertaining. First sentence: "A generating function is a clothesline on which we hang up a sequence of numbers for display." An Introduction to the Analysis of Algorithms by Robert Sedgewick and Philippe Flajolet is another fine introduction.
Recommends awkward · 2021-12 · Mathematics Stack Exchange · on “How many different ways are there to make n dollars with 1, 5, 10, 25, and 50 cent coins.”
The YouTube video Explaining the bizarre pattern in making change for a google dollars [this video] develops an explicit formula for the number of ways to make change for $n$ dollars, using generating functions (but no prior knowledge of generating functions is required). You might check it out.
Recommends awkward · 2022-05 · Mathematics Stack Exchange · on “Coin change problem with specific multiples”
Mathologer solves a similar question (but using a different set of coin values) in the YouTube video Explaining the bizarre pattern in making change for a google dollars [this video] .
Recommends awkward · 2022-09 · Mathematics Stack Exchange · on “Find the general formula for the generating function $g(x) = \frac{1}{1-x} \frac{1}{1-x^5} \frac{1}{1-x^{10}} \frac{1}{1-x^{25}} \frac{1}{1-x^{50}}$”
Mathologer gives a complete solution for the special case where the total amount is a multiple of $100$ cents (which includes one dollar, of course), in a YouTube video: Explaining the bizarre pattern in making change for a google dollars (infinite generating functions) [this video]
Cites chansey · 2022-09 · Mathematics Stack Exchange · on “Find the general formula for the generating function $g(x) = \frac{1}{1-x} \frac{1}{1-x^5} \frac{1}{1-x^{10}} \frac{1}{1-x^{25}} \frac{1}{1-x^{50}}$”
A lot of information. I need some time to digest (e.g. the computation of contributors and quasipolynomial [LINK] ), but I still learned a lot from you answer (e.g. cyclotomic polynomial [LINK] ) . The last suggestion is very similar to Mathologer's method [this video] in the youtube video and Concrete Mathmatics as @awkward mentioned. Thanks!