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==Visual proof==
It would be nice to include the picture proof from http://users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32, since it is clearer than the one already on this page, but I am not sure about the copyright status of the images. Please advise.
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1296+2025 is 3321 (triangle 81)
2025+3025 is 5050 (triangle 100) [[User:KlokkoVanDenBerg|KlokkoVanDenBerg]] ([[User talk:KlokkoVanDenBerg|talk]]) 09:29, 7 April 2022 (UTC)
:It was not removed for lack of proof. It was removed for lack of a published source. Also, 10, 45, 136, etc., are not squared triangular numbers. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 16:14, 7 April 2022 (UTC)
:::They are triangular numbers of squares [[User:KlokkoVanDenBerg|KlokkoVanDenBerg]] ([[User talk:KlokkoVanDenBerg|talk]]) 17:38, 7 April 2022 (UTC)
::::Like how 10 is the 4th triangular number, 45 is the 9th triangular number, 136 is the 16th, and so on. [[User:KlokkoVanDenBerg|KlokkoVanDenBerg]] ([[User talk:KlokkoVanDenBerg|talk]]) 17:40, 7 April 2022 (UTC)
:::::Triangular numbers of squares are a different topic than this article. This article is about squares of triangular numbers. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 17:40, 7 April 2022 (UTC)
::::::I know, but the sum of two consecutive Squared triangular numbers is a Triangular numbers of squares [[User:KlokkoVanDenBerg|KlokkoVanDenBerg]] ([[User talk:KlokkoVanDenBerg|talk]]) 17:42, 7 April 2022 (UTC)
== Better visual proof ==
[[File:Nicomachus_theorem_2D.svg|thumb|Alternative proof using squares <ref id="nelsen">[http://scribd.com/document/758658549/Proofs-without-Words-Exercises-in-Visual-Thinking-Volume-1-Roger-B-Nelsen#page=96 Roger B. Nelsen, ''Proofs Without Words Exercises in Visual Thinking'', Volume 1,] [http://dn720001.ca.archive.org/0/items/proofs-without-words-roger-nelsen/Proofs%20without%20Words%20-%20Roger%20Nelsen_text.pdf page 87]</ref>]]
{{user|David Eppstein}} reverted my addition of this image with the comment ''"we have enough *sourced* visual proofs of this that we do not need to add additional unsourced and illegible ones"''. It was my mistake to not put in a source; I have now found two.<ref id="nelsen" />
This diagram is a more general geometric proof than the other two, which just shows that the relation holds for small integers.
Can you please describe how it is illegible and how it can be improved?
Thanks, '''[[User:cmglee|cmɢʟee]]''' [[User_Talk:cmglee|τaʟκ]] <span style="font-size:80%;">(please add <code>{{ping|cmglee}}</code> to your reply)</span> 00:38, 28 August 2025 (UTC)
:It is a mess of faded colors, formulas, arrows, and upside down superscripted digits. I have no idea what it is supposed to mean. And the whole idea of proofs without words is that they are not supposed to need any text to explain them; your figure is covered all over with text (the formulas and digits). The color combinations are a blatant violation of [[MOS:ACCESSIBILITY]], which is not optional. Only the outline in the central part of it resembles in any way the source image you give, which is clear and simple. All your additions to it compromise the legibility of the image and constitute [[WP:OR]]. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 00:46, 28 August 2025 (UTC)
::{{ping|David Eppstein}} Thanks for the feedback. I tried to explain the source image clearer and make it different so not infringe copyright. If I redrew it as in the original with accessible colours, would it be acceptable? '''[[User:cmglee|cmɢʟee]]''' [[User_Talk:cmglee|τaʟκ]] <span style="font-size:80%;">(please add <code>{{ping|cmglee}}</code> to your reply)</span> 09:49, 28 August 2025 (UTC)
:::Without all the circles and arrows and a paragraph on the back of each one explaining what each one was? Maybe. But is it really different from the existing better image [[:File:Nicomachus theorem 3D.svg]] (with its flat part copied four times, mirrored horizontally and vertically)? —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 17:24, 28 August 2025 (UTC)
::::{{ping|David Eppstein}} The 2D version shows why it is always possible to go from the ''n''th to the (''n'' + 1)th term, while the 3D version (which I drew over 10 years ago, incidentally) doesn't quite show that (I just shows that it's possible for the first five terms). Nevertheless, if it's just a maybe, I can't be assured that it will be time worth spending. Thanks for your attention, '''[[User:cmglee|cmɢʟee]]''' [[User_Talk:cmglee|τaʟκ]] <span style="font-size:80%;">(please add <code>{{ping|cmglee}}</code> to your reply)</span> 22:51, 28 August 2025 (UTC)
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