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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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[[User:CRGreathouse|CRGreathouse]]<small> ([[User talk:CRGreathouse|t]] | [[Special:Contributions/CRGreathouse|c]])</small> 18:22, 31 August 2009 (UTC)
::Nicomachus observes it but doesn't offer a proof. I haven't looked up Aryabhata's discussion, but the general style of the early Indian mathematicians was to state results without proof. In fact this was also common in European algebra down to the 17th century. For example, Descartes stated his 'rule of signs' without anything resembling a proof. I think this was partly because mathematicians often regarded their methods as 'trade secrets', which they could use to solve challenge problems, but also because until the principle of mathematical induction was understood, a lot of problems were difficult to prove rigorously with the available methods.[[Special:Contributions/2A00:23C8:7907:4B01:B180:FF9C:33B1:1040|2A00:23C8:7907:4B01:B180:FF9C:33B1:1040]] ([[User talk:2A00:23C8:7907:4B01:B180:FF9C:33B1:1040|talk]]) 21:29, 9 November 2021 (UTC)Added by same commenter: the passage in Nicomachus is in Chapter 20 of Book 2 of his 'Arithmetic', not in Chapter 20 of the whole work, which would be in Book 1.[[Special:Contributions/2A00:23C8:7907:4B01:5064:8DD0:F21C:FD4B|2A00:23C8:7907:4B01:5064:8DD0:F21C:FD4B]] ([[User talk:2A00:23C8:7907:4B01:5064:8DD0:F21C:FD4B|talk]]) 15:38, 17 November 2021 (UTC)
== Sum of the first "n" cubes - even cubes - odd cubes (geometrical proofs) ==
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:Very interesting, but we're only supposed to put things into the article that are supported by published sources, see [[WP:V]] and [[WP:RS]]. People's own thoughts have no place in the encyclopaedia no matter how good they are, see [[WP:OR]]. [[User:Dmcq|Dmcq]] ([[User talk:Dmcq|talk]]) 20:10, 7 October 2013 (UTC)
::This my work is currently being published on italian magazine. <small><span class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Ancora Luciano|Ancora Luciano]] ([[User talk:Ancora Luciano|talk]] • [[Special:Contributions/Ancora Luciano|contribs]]) 08:39, 9 October 2013 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
:::Honestly, the evaluation of these limits is a standard exercise of calculus. No more reasons to be included in any encyclopaedia, than the evaluation of 129+934. It's a computation. Just apply the definition of Riemann integral for x<sup>k</sup>. [[User:PMajer|pm]][[User talk:PMajer|<span style="color:blue;">a</span>]] 00:22, 19 November 2013 (UTC)
== Probabilistic Interpretation? ==
The LHS indeed is correct, but the RHS doesn't make any sense. <small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/1.162.222.80|1.162.222.80]] ([[User talk:1.162.222.80|talk]]) 07:39, 16 April 2016 (UTC)</small><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
:Indeed, both probabilities coincide with the left and right side of the Nicomachus identity, but the events are certainly not equivalent. In fact, one can only conclude their probabilities are equal ''because'' of the Nicomachus identity. There are no further references either, and I could not find anything else in google. I think this is an original idea someone posted. <!-- Template:Unsigned IP --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/61.127.95.156|61.127.95.156]] ([[User talk:61.127.95.156#top|talk]]) 13:08, 12 February 2018 (UTC)</small> <!--Autosigned by SineBot-->
:I agree; that section is at best confusing, and at worst wrong. I've added a "Confusing section" tag. [[User:PatricKiwi|PatricKiwi]] ([[User talk:PatricKiwi|talk]]) 06:54, 16 February 2021 (UTC)
::It needs a source, but the RHS is obviously just the Cartesian product of two isosceles right triangles 0≤x≤y≤N and 0≤z≤w≤N in the xy and zw planes, a natural geometric interpretation of a squared triangular number. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 07:01, 16 February 2021 (UTC)
:::Thank you for this explanation; it's a lot clearer than the paragraph in the article itself. Perhaps you'd like to improve that paragraph - or at least give it a tag that you feel is more appropriate than "Confusing"? [[User:PatricKiwi|PatricKiwi]] ([[User talk:PatricKiwi|talk]]) 07:28, 16 February 2021 (UTC)
==Triangular numbers of squares==
I recently added that these numbers are the alternating sums of Triangular numbers of squares, similar to how triangular numbers are the alternating sums of squares, however this was removed because of a lack of proof:
0+1 is 1
1+9 is 10 ( triangle 4)
9+36 is 45 (triangle 9)
36+100 is 136 (triangle 16)
100+225 is 325 (triangle 25)
225+441 is 666 (triangle 36)
441+784 is 1225 (triangle 49)
784+1296 is 2080 (triangle 64)
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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