A Proof - With Updates...
That a multi-digit number in Base 10 ending in "1" is less likely to be prime (plus some hexagon considerations)
UPDATE: Edit - tis December 6, 2023 - tis 1047 as I type this - here is a snippet you might appreciate - I know I do.
I think if I ever changed my long-time “icon”, my “moniker” I’ve used to present - “who I am” - Buffalo_Ken - for so many years now and I’ve shown the origin image of it - anyhow, if I ever changed it - I would change it to that image above - what you think it is?
So - since I’ve typed this I might as well provide a link to the place mentioned - it has the original image on it of who I am.
http://www.carbon-tax.us/
You've got to read down to get to the image
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UPDATE: Edit - tis December 3, 2023 - 12323 and tis 1314 in minute time - so if you want to know what a “tight” space is of pieces connected, then take the 23 shapes with 5 “perfect” hexagons below. That gives what? One hundred fifteen (115) shapes in total (23 * 5) and if a hexagon individual is characterized as “perfect”, then please be aware it is imaginary in a way cause nothing - not even nothing - is perfect. Regardless, going with that premise, each hexagon has a fixed area 2-dimensionally and so the total area when wrapping around the perimeter of any collective shape with all twenty three (23) five-tile-hexagon shapes connected has a lower bound = 23 * 5 * area of a perfect hexagon individual. If you combine the shapes in said manner with no space betwixt, then that would be a top score in the “game” I’m imagining now….so…I thought I’d share that idea. Feedback is desired.
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UPDATE: Edit - tis December 2, 2023 - 12223 (or 12’223 for “purist”) as I like to state it….and here are all 23 shapes - please tell me if you see one that is NOT discrete.
Shape that was there has been temporarily deleted - proof checking in progress
tis 122323 1919
I’m sure there is a way “easier” to present this and as I’m working on shapes with six (6) tiles each I’ll try to figure that out for the sake of good communication. With that said, no matter how one flips or twists the above shapes - each one is discrete and distinct 2-dimensionally as far as I know assuming I didn’t make a mistake. There is some math buried in this I know.
Now with that just posted above you tell me this - if you took each shape and rotated it around how “tight” could you get the pieces to fit together?
I suspect the “white space” inbetween could be reduced by about 90% if not completely - that is a puzzle in and of itself - are you up for the challenge?
BK, 12223 1641
Consider this - if each shape was a “card” in the deck and they were picked randomly could the players place the pieces to fit together tightly? And if they did and there were no blank spaces in-betwixt, then that means a high score in the game per one way of scoring things…
Can you imagine this - it could be both solitaire or group play!
There are so many options for how this could be a way to learn about shapes natural.
I think I might try to do this for myself and when I do I’ll come back here and present the image tight…..if I do…but looking at the shapes just now and imagining the possibilities - I’m pretty sure they can be connected without any space in be-twixt….but there is some luck to it depending on the “rules” of the game - my initial thought is - pick the pieces one a time randomly and connect them in a manner facilitating a “tight” space - and another question mathematical is - could it be no matter the order the shapes picked that with “proper placement” there will be no space in-betwixt?
I have no idea what the answer is!
Sometimes you might as well finish what you start - here are the final shapes I think when 5 hexagons are connected 2-dimensionally - there are five “new” shapes in these images as I’ve gone back in forth making corrections and if I got my shapes correct finally - that gives a total of 23.
Please point out any mistakes if you see them, but I’m pretty sure 23 is the number of shapes with 5 hexagons independent of orientation and now that this is down here and published - I reckon tis time to move on to the possibilities when six hexagon tiles are connected….
(a little edit here - 122523 1514 - I’ll never tell unless you ask!)
Tis presently 112523 and tis 7:48 pm and I’m done for today…
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OK, I decided to try, I’ll come back and check for mistakes later, but here is the next two shapes on the way to 23 total:
This can be trick avoiding redundancy, but I’ll come back and double-check…
Last edit for today - tis 1843 - this is ridiculous, I’m correcting what I got correct in the first place - must mean tis time to take a little rest on this…
I’m fond of “Shape 4” - it ain’t its fault it came later in diagnosis, but sure enough it is redundant with shape 2c below.
other corrections have been made I believe - but I’ll double check again…
(122323 1928)
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OK, so after making it through 4 of the shapes made with 4 tiles, the number of shapes discrete with 5 tiles is:
4 + 10 + 4 + 0 when adding it up with those four starting shapes accordingly. So that gives 18 shapes so far - there are five more to be discovered assuming I haven’t made any errors, but I’ll come back here and double-check my work and get the numbers accurate and provable. That 4th shape - I wasn’t sure for awhile and I’m still not totally sure, but it helps to practice and this methodology is easily proven - in the real world - the macro world - the world we all can sense. That in contrast to “quantum physics land of imagination” has beauty easily understood in my mind.
OK - c u later…
112523 1851
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Here is the next set of hexagon shapes connected five tiles.
(Edit - 122323 1917 - I just found an error in the image that was below - here is a “correct” version - but now I got some other things to “double-check”)
I know I might have some typos or there could be mistakes, but after I get through doing all of this, I’m gonna post an article with the proof up to six hexagons - and it will be double-checked no doubt and accurate and my guess is the total number will be prime.
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Here is the latest update: 112523 1329
Here is the proof (see below):
(Edit tis November 24, 2023 or 112423 per nomenclature I prefer - more detail on the hexagon question discussed below is presented in this image that shows the 7 possible discrete 2-dimensional images that can be made with hexagons independent of orientation - it is a fixed number and anybody can figure this out on their own - that in a way is the beauty of it I think…)
Do you see - there are four (4) hexagons connected in each combined shape and there are a total of seven (7) discrete shapes total 2-dimensionally. You can ignore the images within the shapes, but they make for a good game potentially - the main question I have is how many discrete shapes are there when there are eight (8) hexagons in total connected 2-dimensionally. When there are five (5) hexagons, the number of shapes is 23. I’ll solve it for six (6) hexagons soon enough.
Meanwhile, I’m becoming an advocate for propane as a good fuel for those who choose to be self-sufficient.
It was posted per the following time stamp:
2023 - 11 - 12T20:10:50Z
This was about 9 days ago….so must of been
111223 1450 per my “nomenclature” - November 12 was “9 days ago” was it not? I just snipped the image above - so you figure it out. It came from here:
https://www.sott.net/article/157471-Stonehenge-builders-had-geometry-skills-to-rival-Pythagoras
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You know there are some of us love our children more than you could ever imagine assuming you don’t love your children like we do.
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Below is the 2nd heater I got setup today:
Propane heat - easy to setup for the everyday man. It fits in here at our getaway place in the forested hills by the river.
~~~~~~~~~~ Edit - 112523 1039 ~~~~~~~~~~
OK, it is November 25, 2023 10:39 am (give or take) Eastern US time, and I am having fun presenting some harmless mathematical ideas I’ve been “stewing” over for quite some time - especially with respect to the “hexagon puzzle” (see Comments below for additional detail). So, with my recognition yesterday that I could simply present images using tools I already have hand, below is the first image showing how the number of discrete shapes (independent of orientation) with five “perfect hexagons” connected can be determined. It starts with one of the shapes presented above when there were four connected hexagons and then presents the possible “new” shapes that can be made when adding a 5th tile. So if you recall, there were a total of 7 shapes with 4 tiles connected (per the image at the top of the article), and I will work through each of them and show the “new” shapes when there a 5th tile is connected (green fill). It is important to remember previous shapes already created so that only discrete shapes get counted.
More to follow…..although I may just go ahead and dedicate a post to this hexagon puzzle so as not to get too mixed up with the prime number considerations….
This now 10:50 am.
~~~~~ Edit - English Heritage Scarf for Sale - ~~~~~~~~
Get it here: https://www.english-heritageshop.org.uk/mosaic-scarf-viscose-65-x-180cm
Stonehenge is a place of geometry no doubt.
BK
This is what is on my mind…..
* ??? *
Well, here is the proof why any whole number ending in digit #9 is less likely to be prime versus whole numbers ending in digit #3 or digit #7 in base10. Assuming you have read the article, R U Ready?
1. 3*7=7*3=21. 3*9=9*3=27. 7*9=9*7=63. So, giving these each a “count of 1” versus counting them both ways, gives one ending in #1, one ending in #7, and one ending in #3, respectively.
2. The other possibilities are 3 * 3 = 9, 7*7 = 49, and 9*9=81. This gives 2 ending in #9 and one ending in #1.
3. Therefore, per the approach of not “double-counting” forwards and backwards, the totals are two ending in #1 and two ending in #9 versus just one ending in #7 and #3, respectively.
4. Therefore, whole numbers ending in #9 are less likely to be prime versus those ending in either #7 or #3, respectively.
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BK
* If somebody wants to prove that incorrect, please do.
I might as well go on record with the post above linked - I'll link it below cause it was here I posed this on 101822 1430:
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OK, so I did a search using the term "geometry" and this was the top article so this is where I'm gonna try in my mind to take the hexagonal puzzle in imagination mostly out to the 4th tile. Wish me luck because this is all a mind game just now mainly, but later if I'm still alive, I'll take the hexagon tiles I have and put it to a real-world test. Probably just moments after posting this.
~~~~here goes~~~~~~
1. First tile is placed - there is one shape only distinct.
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2. Second tile is placed - there is still only one shape distinct, because two tile edges connecting on hexagons have no differentiation without consideration for orientation. Basically, the two of them together will be the same regardless of which two sides initially connect no matter perspective 2-dimensionally - the shape is singular even with two pieces. Can you visualize this? I'm not talking trash - really.
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3. As explained elsewhere, and per the logic expressed already, there are 3 possible shapes without consideration for orientation and in a 2-dimensional plane or field when 3 hexagonal shapes are connected - side by side. One of the three is three in a row so that they form a line (***). The second of the three is three combined so that they form the beginnings of a circle so-to-speak - they are all matched together in nice symmetry with each other (*$*). The third is when one of the many options to place the 3rd tile after two already been placed off to the side of either starting tile not being either the straight line option (#1) nor the all packed in there option (#2), but at the end of the day, there is only one discrete shape and it doesn't matter really which side gets connected even if there are 4 options for this on the two previous played tiles having 2 each....by the way, for the 3rd tile placement into a discrete shape there are two options for #1 above and two options for #2 above as well, but for the 3rd shape there are 4 options and so likely its occurrence might be higher all other things being equal.
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4. Herein lies the formula already I suspect, but it wouldn't be difficult to extend this out to the fourth hexagonal tile placements, just walk through it per each of the 3 options indicated above. For each of the 3 there will then be several options I suspect when the 4th tile is added. This is where doing it physically is helpful because when unbiased a mind sometimes has a hard time coming to conclusion with respect to redundance without orientation - it helps to get the senses involved.
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5. If you can do it for 4, you can do the same thing for 5. I think the solution beckons and I'll figure it out myself if need be.
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Peace and I hope all are doing well and how about them damn Buffalo Bills?
Nobody wants to play them and now they get a week off to relish, but I hope they don't relish too much.
The season is young.
Peace, Poem of the Day, 101822 1430
BK
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If you got the solution - please don't tell me - I want to figure it out on my own.
Oh yeah - here is the link:
https://www.sott.net/article/157471-Stonehenge-builders-had-geometry-skills-to-rival-Pythagoras