Data Science Wizardry Blog by Attila Vajda

Proofs that count

Proofs that count #

Reading the Proofs That Really Count book furthers ability to read mathetematical notation. It is an interesting book. The Fibonacci sequence can be written combintorially. Wondering how to understand the multiplication of binomial efficients, reading a combinatorial proofs seems to provide ample exercise in thinking in terms of combinations.

Prove each of the identities below by a direct combinatorial argument.
Identity 12 For n ≥ 1, f1 + f3 + · · · + f2n−1 = f2n − 1.

Now this is something I struggle with!

After being very frustrated, I made some insight! 😌

Understanding how n-boards grow can be challenging. Putting it into words contributes to seeing it a bit more clearer, by making effort to assemble the puzzle pieces. It is challenging to figure out

The meaning of $f_k$ was a mystery until plugging in values, and colorcoding corresponding n-board tilings. fk, the number of tilings that have a domino as a last bit Plugging in 3 Plugging in 4

I realised $f_k$ refers to cardinality of n-board tilings on $k$ cells.

This drawing, then brought an "Aha!" moment, when I saw that by putting a domino on the last two cells, the first three cells remain for combinations: Last two cells are occupied by a domino

Journal pages of mathematic reasoning, with Kanji and Hiragana learning Journal pages of mathematic reasoning, with Kanji and Hiragana learning

A little free form exploration #

At the end of last summer I became very excited when I read Measurement and Mathematician's Lament by Paul Lockhart, so I started a free form exploration with a compass and straightedge, looking for patterns. Yesterday I came accross a related problem, and revisited these pages. It is delighting to find, and wonder about the ratio of a square, and a circle.

Journal pages of maths exploration, relationships of circle, square, pi

A square drawn over a circle, as a fraction

When I first found this I was happy, but I wasn't sure that it was meaningful (not sure what the right word is). When it worked as a solution to a problem, and the numbers were making sense, it seemed really cool!

Square to circle fraction, and area of square to area of circle as fraction, equals 4/π

>>> 1/(math.pi*(1/4))
1.2732395447351628
>>> 4/math.pi
1.2732395447351628

Area of quarter square to quarter circle, drawn as fraction, equals 4/π

"This is crazy!" is what it makes me think. It's fun, it makes me smile.

Square pie. Circle in square, with Pi written in the circle, as pi is the area of a 2by2 square circle

Charcoal drawing #

Spending a few minutes drawing with charcoal can be fun as well. Charcoal drawing of hand

Wow, inverting this image gave it a night sky feel! It is drawn on wrapping paper.