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Thanks Bob for all that effort. I'm glad you understood what I was trying say. The piano analogy was the best way I could think of describing the query although, of course, there were many other options. Did having a piano at your side actually help? Was that just a little bit of luck?
I normally have very little interest in anything mathematical. Number theory, equations, mathematical formulas are things I left behind me in my school days and they were a long time ago. This puzzle just came to me out of the blue when I was laying in bed and staring at the slats of the window shutters and began counting them up and down til I noticed what was happening. I needed to know a little more about it.
So, your responses seem to confirm what I was seeing but you did that using mathematics to provide you with what you, I think, call Complete Proof. A process which, I imagine, is essential to uphold the theory or the working out of the problem.
I think I still have a couple of unanswered questions though. The first I think I probably already know the answer to. And that is, as I mentioned in my first post, do these patterns continue ad infinitum? The example I mentioned was if you played a forty note tune, would you only have to go back and forth forty times but for a forty-one note tune have to do so eighty-one times (41 x 2 = 82 - 1 = 81)?
I imagine that's correct which, to me, is a simple but quite amazing fact. To know with certainty beforehand that if you played a 3,027 note tune, to get to the point of repeating the initial tune, you'd need to play the sequence through 6,053 times (3,027 x 2 = 6,054 - 1 = 6,053), is sort of "cool" as they say these days. But, I suppose, you'd need complete proof of that too to be absolutely confident that it is actually correct. So, has that ever been done?
Which brings me to my second unanswered question. Has all this been recognized before? The answer must be yes because I'm 1000 % certain I'm not the first person to have queried this over the past several thousand years of mathematical problem solving. So, if the two "phenomena" (the odd one and the even one) have been recognized before, someone must have given them names, or whatever the equivalent would be in mathematical terms. I can't imagine, after all this time, the only way to ask another person about this would be in the long-winded, analogy-finding way I just did with you and for you to then have to go through all the effort you went through to research and prove the logistics of it. Someone must have done all that a very long time ago and put it on the record books as a well identified problem with an actual formulaic solution and gave it a moniker. We two can't be the first. If so, we'd give the whole process a name so that anyone else after us would instantly know what we're talking about.
So, do you know, or are you able to find out, if there is an established term or name or written descriptive of any sort for what we've just "discovered"?
I hope you don't mind me asking and I do thank you for time. I imagine you probably found it quite interesting to work on.
Hi Bob and thanks fo
Hi Bob and thanks for your reply.
Yes, I can understand that it might be difficult to follow the sequence initially but if you do try it out on your piano a few times I'm sure you'll see what I mean. Especially so if you listen to the first three notes and wait for them to be repeated. 212 follows 123 because you've just gone from left to right (123) then continued but by going immediately right to left and then back again (212). From there all you're doing is "rolling over" the sequence until you're back to the original three note tune. I can only suggest that you try it a few times until it makes sense. Using a piano isn't the only way to investigate this but it does have the advantage of giving you the musical notes to bring it together.
I Imagine someone wi
I Imagine someone will know the name of this mathematical sequence and how it progresses the further along the sequence you take it. I have no idea about either but I would like to know.
I'll use an example of a piano keyboard to hopefully describe the sequence as best I can.
If you imagine starting on the extreme left of the keyboard and play, left to right, keys 1 and 2 it will play a short tune. Do this again and will repeat the same tune. Now play keys 1, 2 and 3 to give another (three note) tune. Now we have to go backwards to play keys 2, 1 and 2 followed by 3, 2 and 1 followed by 2, 3 and 2 followed by 1, 2 and 3 at which point you've just repeated the initial tune. So it's a continuous, rolling, back and forth sequence which finally brings you back to the tune you first played.
You will have noticed that to repeat a three note tune you had to go back and forth five times to get you back to the initial tune.
If you then apply the same continuous, rolling, back and forth method to playing a four note tune it would go like this..... 1, 2, 3, 4 then 3, 2, 1, 2 then 3, 4, 3, 2 and finally back to 1, 2, 3, 4 at which point you've just repeated the initial four note tune. But this time you only had to go back and forth four times to repeat the initial four note tune. So fewer times than you did for the three note tune.
Expanding these sequences to a five note tune will see you going back and forth nine times but for a six note tune only six times. For a seven note tune it will be fifteen times but for an eight note tune only eight times.
So, so far, it appears that for tunes with an even amount of notes you will go back and forth the same amount of times as the number of notes in the initial tune. Four times for four notes, six times for six notes, eight times for eight notes etc, etc. For a tune with an uneven amount of notes there seems to be a formula you can use to calculate how many times you will have to go back and forth to get to the point where you are back to the original tune. At these shortish tunes at least, if you multiply the amount of notes in the original tune by 2 then subtract that total by 1 you get your answer. So for a three note tune, multiply by 2 gives you 6, minus 1 gives you 5. Which is the amount of times you had to go back and forth. For a five note tune multiply by 2 gives you 10 minus 1 gives you 9 which is the amount of times you have to go back and forth to get back to the original nine note tune. I Imagine this formula might continue ad infinitum for both even and uneven numbers but I might be wrong. That would seem too simple. If, for example, if you played a forty note tune, would you only have to go back and forth forty times but for a forty-one note tune have to do so eighty-one times (41 x 2 = 82 - 1 = 81)?
If I've explained this all clearly enough and you understand my query, is there a mathematical name or formula for this and how does the formula evolve ad infinitum?
Thanks in advance for any responses.
Frank
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