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While trying to solve a chiming clock problem algebraically, I came across a solution which used "Rainbow" Addition Facts.

Problem

An old chime clock strikes one chime at 1 o’clock, two chimes at 2 o’clock, three chimes at 3 o’clock and so on. How many chimes will it strike in a 12-hour cycle?

Solution

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To be honest, It's my first encounter with "Rainbow" Addition Facts and it looked so alien to me at first.

Couldn't find any information regarding "Rainbow" Addition Facts on Internet other than couple YouTube videos which shows how to use them.

I'd hear about history of "Rainbow" Addition Facts and theory behind it.

codespeare
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    Your question is better suited at History of Science and Mathematics SE or Maths Educators SE. – Toby Mak Oct 04 '20 at 05:37
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    what are "Rainbow" Addition Facts? – Misha Lavrov Oct 04 '20 at 05:39
  • @TobyMak, I didn't know existence of such specific meta QAs. I searched here on this Math meta for "History" keyword and found lot of questions. I thought it's okay to ask. Again, I welcome if someone can merge this Question to appropriate place – codespeare Oct 04 '20 at 05:40
  • You can always delete your question and ask on one of these sites. It would be good if you can explain what they are and give an example, such as $1 + 9 = 2 + 8 = 3 + 7 = 10$. – Toby Mak Oct 04 '20 at 05:42
  • @Misha Lavrov Complicated but only handful resources out there with information about it. It does the job nicely but I don't have enough confidence to use it in practice. – codespeare Oct 04 '20 at 05:42
  • @TobyMak Please see my edit. – codespeare Oct 04 '20 at 05:46
  • That's much better than before. It seems like you want to know why $1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}$: there are plenty of resources that address this online (such as a picture proof with stones). – Toby Mak Oct 04 '20 at 05:50
  • Similar to Gauss' story from his childhood to add up $1+2+3+\cdots+100$. – cosmo5 Oct 04 '20 at 05:52
  • I'm voting to close this question as an abstract duplicate of Prove $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$. – Toby Mak Oct 04 '20 at 05:55
  • @TobyMak It doesn't. I need to understand the actual theory behind Rainbow facts. I know your answer as a solution but I really need to understand how Rainbow Facts works by finding common number and getting an answer by multiplying that common number. For above case, the common number is 13 and it get solution multiplying by 6 which is 78. – codespeare Oct 04 '20 at 05:57
  • This is no different to Gauss's proof (the first answer in the link), where you write the sum backwards and add it to itself, as $1 + n = 2 + (n - 1) = 3 + (n - 2) = \cdots = n + 1$ all add to the same number. The label "rainbow facts" is just a distraction. – Toby Mak Oct 04 '20 at 05:59
  • @TobyMak I guess it does but I am looking for more of layman explanation because my head is not suitable for number based proofs. I'm so sorry! – codespeare Oct 04 '20 at 06:07
  • It's okay. Try to convince yourself of Gauss's proof by trying it with small numbers, such as $4$. Write $1 + 2 + 3 + 4$ and add it to $4 + 3 + 2 + 1$. Complete the proof, and then try it with another number. – Toby Mak Oct 04 '20 at 06:09

1 Answers1

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I've never heard of "rainbow facts", but it seems to be a very roundabout way of saying that $1+2+3+\cdots+12=78.$

In general, $$1+2+3+\cdots+n = \frac{n(n+1)}2,$$

which you can prove by writing the sum twice and pairing elements that add to $n+1$.

Théophile
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  • Similar to Gauss' story from his childhood to add up $1+2+3+⋯+100$. – cosmo5 Oct 04 '20 at 05:54
  • @Théophile I think Rainbow facts does it differently. It find a common number and multiply that common number. For above example, The common number is 13 and it multiply 13 x 6 which gives 78 – codespeare Oct 04 '20 at 05:54
  • @codespeare Yes. It finds the common sum which is $n+1$ and multiplies it with $n/2$. This is not different from what this answer tells. For your example, $n+1=13$ and $n/2=6$. – cosmo5 Oct 04 '20 at 08:53