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Yang Hui’s Triangle Hides Fast Algorithm – The Traditional Chinese Approach

Audience:

We use the hidden summation formulas within Yang Hui’s triangle to deduce the formula for the sum of higher integer powers of the numbers from 1 to n. We also demonstrate how the successive difference method can accelerate these calculations. This method first appeared in Yuan Dynasty Zhu Shijie’s book “Precious Mirror of the Four Elements” and was proved by modern Chinese mathematician Hua Luogeng. Lastly, we briefly review an application of this summation formula to compute a definite integral.

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5.84 Overall score*

82 Rank

9 Votes

4 Comments

Comments

7.8

Nice for students and teachers. I learned something new too. Thank you, don’t stop making contents!

7.5

Nice video, the piano music is very annoying because of volume and complex melody which also doesnt help staying focused. Nice transition from relatively obvious/broadly known to the unknown/new insights. Very fitting for the category.

Yes, although the video was basically like someone writing on the whiteboard with a lot of text.

I really this video. It’s a clear explanation of a foundational combinatorial topic, with a historical context that I think the average viewer won’t have heard before. But beyond any technical praise I could give, I just had fun watching it. In particular, your aesthetic voice comes through very strongly through choices that you made in animations (the theme of underlying binomial coefficients, the color coding when we get to Zhu’s theorem, etc.)

[Fair warning— I am biased because combinatorics is my area of expertise :3]

My biggest complaint about the video is that you hint at the idea that this method for finding sums of squares and cubes can be generalized. This is of course true, but to go from the system-of-equations algorithm to a “general case” closed formula has subtleties (honestly I can’t remember if it’s even possible; Stirling numbers), and I think that would be worth pointing out explicitly.

A small aside, I think it was a great choice to not get distracted by the linear algebra for that system of equations. There’s certainly an interesting story there but for your purposes, no need to introduce that terminology and the video is tighter for it.

In fact, I see subtle choices like this made all throughout, clearly you worked hard to refine the script to say only what you needed to support your thesis. Very commendable; easy to get distracted with so many interesting connections nearby.