Data Science Wizardry Blog by Attila Vajda

Lagrimas negras

Keeping learning combinatorial proofs #

Joy of of X by Steve Strogatz is a cool book!

I didn't know 2+5+8=

$For\ {n ≥ 0}, f_0 + f_2 + f_4 + · · · + f_{2n} = f_{2n+1}$.

Haha, I was befuddled at why summing every even numbered sets of tilings, the sum would not be $f_{2n+1}$. Then I realised the 2n+1 stood for odd index sets!

$f_{n+2}+f_{n-2}$ is the number of tilings of $3f_n$, the sum of two cardinalities, but/and also the union of two sets + ⋂ v. This is an insight, I was surprised to read the author use or, because it showed another connection.

Every paragraph, even sentence / notation, there is insight, this is so interesting! $3f_n = f_{n+2}+f_{n-2}$ the 1-3 correspondence is there is one cell for each cell of the $f_n$ tiling Writing it down, I learn to see accurately what is happening: For each n-tilings there are three (n+2) OR (n-2) tilings. I love this!, how the after finding a small insight, I try fitting/seeing the notation piece with its newfound meaning - like how, for example, the n-tilings, and the OR statement are expressed in the original statement. The question, and answers the authors are making are the translations of the notation. Fantastic! The answer to my question of A-s and B-s being appended to the patterns, is written down

I CAN read books like Proofs the Count, it's awesome! Reading, like reading Spanish, and learning through exposing myself to the material, learning the language, exponentially! "This kind of reading is definitely not the passive kind of reading, this reading is really cool!"

What happens is that I read, I don't understand the notation, nor the new vocabulary, and I try to recall, mentally visualise the examples I created, and previous examples that are given. Using my fingers I check the pattern, and I get some answer. Vague images start to form, and reading on, struggling to fit the pieces together little bits of insight emerge, that are always surprising, and very satisfying. This is a fun experience!

I plan to read, and reread this book.

I don't know why this is, but I just love this! I reread the paragraphs, and gain some more insight.

You can go a bit lighter on moving on

It is surprising, again, and again. $2^{n-2-k}$ was unknown, even a bit intimidating, and now it makes sense. $n$ is the n-board's length, $-2$ is the two zeros in the unchanging 00, and $-k$ is the unchanging k-board part. We raise 2 to this power, to get the variations of the changing part after 00. The first occurrence of 00 is enough to indicate a non-board.

Jazz chord combinatorics #

All of the jazz chords are combinations of the 12 notes! They can be sounded on a single octve of the piano

Combinatorial proof learning binomial coefficient #

strategy to spend some time and effort on a translation/problem, and then read on, and have "4-5" problems to think about, intervallically revisit problems (fibonacci spiral?), and read through the book, instead of banging my head against a sole problem I don't comprehend just yet

It feels like learning a language. Speaking mathematics.

It is very interesting, how in $f_{n}$ $n$ means the board length and the index of the set of tilings.

Similarity with logarithms and their square inverses.

Proofs are often much simpler than I would expect, or fear them to be. I am sometimes anxious about some complex solution to be found, and it turns out to be simple. Does my expectation of complexity hinders the learning?

Following up intuitive strands of thought, looking at concrete examples can be good enough, valuable, to learn notation, to get a little bit of more insight into their meaning, and connection. What does $f_{m+n} = {f_m}{f_n} + {f_{m-1}}{f_{n-1}}$ mean? You can look at the tilings and recognise the first few examples. There might be a faint, doubtful urge to discard doing this idea, of looking at how the pattern appears in the combinations, because it seems too trivial.

After collecting these notes, it is easy to just copy and paste them into the blog article.