A 3D #MathGIF proof without words for the sum of the first odd numbers :
1+3+5+...+(2n-1)=n²pic.twitter.com/rJk0sjIzLq
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Which makes me think that a surface-of-a-tetrahedron number would be the sum of two square numbers. (Had to delete this tweet and check before tweeting again! Seems to work.)https://twitter.com/panlepan/status/1215989307854422016 …
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Alternatively :) 1+3+5+...+(2n-1) =(1+2+3+...+n)+(1+2+...+n-1) by separating 3=2+1, 5=3+2,... =n(n+1)/2+n(n-1)/2 =(n+1+n-1)n/2 =(n/2)*2n =n²
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Nice. How about, by adding n 1's and subtracting n, 1+3+5+...+(2n-1) = (2+4+6+...+2n) - n = 2(1+2+3+...+n) - n = 2n(n+1)/2 - n = n² + n - n = n²
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Yeah......I'm not getting that
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You have to look at all the dots, not just the white ones. The change of color is there to signal the change when you move from one odd number to the next one. Does that make sense now?
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Whoever claims that they understood this, is obligated to explain it to the rest of us. Will ya?
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Challenge accepted: the diagram illustrates that you can break up (technically "partition") n² dots into a set of (flipped) L-shaped groups of dots respectively containing 1, 3, 5, ... dots all the way to (2n-1). The diagram only goes as far as n=6 but makes the pattern obvious.
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What is the use of this?
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