Summing Odds via Tokens @MathVisualProofs

Uploaded September 2026 | Updated September 2026, 2 weeks ago

Summing Odds via Tokens

This is a short, animated visual proof showing one method to find the sum of the first n odd positive integers by stacking tokens in a square array of cells.

If you like this video, consider subscribing to the channel or consider buying me a coffee: buymeacoffee.com/VisualProofs. Thanks!

To see a narrated version of this video, check out:
youtu.be/jq5AYCgkciE?si=h8Y1h7h62Cr9WYPF

To see the other methods for finding the sum odd integers visually, check out the following videos:
youtu.be/I4A2T55k0cg
youtu.be/x3qfFBNRRDg
youtu.be/ZeEOgbLo0Rg
youtu.be/6H6nCpGW1nU

For other sums related to odd integers, check out these visuals:
youtu.be/WKxoi2i0EaE (summing odd squares)
studio.youtube.com/video/9BC1klE8jMk/edit (alternating odd sums)
youtu.be/MmOTqrtbtFQ (odd sum ratios)
youtu.be/fTBvVeURb3Q (odd sum ratios II)

This animation is based on a visual proof due to Timothée Duval that appeared in the Mars-Avril 2007 issue of Tangente no 115 (link. page 10).

You can also find this diagram in Roger Nelsen's book on page 114: (I may receive a small commission at no cost to you for this affiliate link): amzn.to/47kfG7w

#manim #math #proofwithoutwords #visualproof #squares #oddnumbers

To learn more about animating with manim, check out:
https://manim.community

$Self-Similar Geometric Series Sum Self-similar geometric series sum This is a short, animated visual proof demonstrating the infinite sum of the powers of 1/2 using a square. To see a longer version with explanation, check out: https://youtu.be/AiznRQPIwTs If you like this video, consider subscribing to the channel or consider buying me a coffee: https://www.buymeacoffee.com/VisualProofs. Thanks! If you like this type of video, here is my geometric sum playlist: https://www.youtube.com/playlist?list=PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw This dissection is loosely based on a proof by Roger B. Nelsen from the February 2006 issue of Mathematics Magazine page 60. (https://doi.org/10.2307/27642904). An animated version of that exact dissection is here: https://youtube.com/shorts/RoovN01q83Y or https://youtu.be/saJJGNlsfn8 #mathematics #mathvideo #math #calculus #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #geometricseries #series #infinitesums #infiniteseries #selfsimilarity #fractal To learn more about animating with manim, check out: https://manim.community$

$Koch Snowflake: Infinite Perimeter but finite area This is a short, animation showing the Koch snowflake and discussing the fact that it has infinite perimeter but encloses a finite area. #mathshorts #mathvideo #math #fractal #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #limits #mathematics #perimeter #kochsnowflake This is a standard result that can be found in numerous places, such as Wikipedia. To learn more about animating with manim, check out: https://manim.community$