The Hexagon Puzzle - Six Tiles

One puzzle finished leads to another - This is my end of year puzzle

Before we get “serious” - I would like to share the purest image of “pink” I’m aware of - and I’ve done some research on this - so please don’t ridicule this image of pure pink in my mind - beautiful beyond words:

If you disagree I understand - but know this - I had the full experience of being there in the moment - and that informs my sentiment.

OK - TIME TO GET SERIOUS!

(Update - tis 5224 - I just found a mistake - and twas in a correction - sometimes tis true you try to correct too much - you make errors when you do - so the correction is below as best I know how to communicate as such - but the numbers on this I reckon remain in flux - and I been seeking out others to help “double-check” this harmless puzzle but so far most don’t really seem engaged or willing to assist in this effort to find the fixed number of shapes - that if one has a mind full of geometry is irrefutable - I sense patterns in so many things)

Tis 121723 1153 (December 17, 2023 - 11:53 am where I reside presently in Charlotte, NC) and I have figured out my methodology for presenting the solution to this puzzle - (see links below for more detail).

So with no further discussion here are the initial images that will lead to the solution I intend on finding soon enough:

Shape 1 is above - it starts out with “5 in a row” as the base shape. It gives 4 new shapes with six (6) hexagons (connected 2-dimensionally).

Shape 2 is below - it starts with “4 in a row with a 5th off to the side downward”. It gives 12 new shapes with six (6) hexagons.

(this is a “sneaky” comment - tis 5224 2104 - not smarky - sneaky mind ya - speaks to what I have learned and how it going to make the double-checking so much easier - just look for the “base” shapes already addressed in the newer shapes - and if you see such - tis redundant already by definition - so I’m going to follow the logic of this and being I truly have these shapes in my mind and on the table - I’ll double check this - the number may change, but I hold out some conviction that it will still be prime - but I don’t mind getting proven wrong - let the fact of the matter speak for itself - cause the fact of the matter is there is a finite definitive whole number of shapes answer to this puzzle of mine in Base10 math - and that truly is the beauty of it - there is no uncertainty about it - but does anybody else know the answer - and has this line of thought experiment been considered before? If not - that speaks to the beauty of learning I put forth - so maybe this is not really sneaky - and if you have actually read this - then maybe you ought post a comment - cause how else am I gonna know if I have friends “out there” - how else is this to be known?)

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Edit: 122123 741

Shape 3 is below - it starts with “4 in a row with a 5th off to the side upward”. It gives 7 new shapes with six (6) hexagons.

(I should acknowledge that part of the beauty of working on this puzzle in so many different ways is that eventually “errors” are identified - and I realize now in an earlier post I missed one of the shapes above - so I’m going to correct this).

Shape 4 is below - it starts with “4 in a row with a 5th off to the in the middle”. It gives 2 new shapes with six (6) hexagons.

(Image deleted - twas a mistake - 5224)

Now - that completes all of the shapes that have “four in row” - thus any possible “new” shape for consideration later on resulting in “four in row” is most likely, if not definitively, redundant with one of the shapes indicated above (Shape 1 - Shape 4).

I’m working my way through this - I had a bit of a “cold” the last few days that set me back, but I still fully intend on going through all of the 23 shapes that can be formed with five (5) hexagons connected to determine the total number of discrete shapes that can be formed with six (6) hexagons connected.

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The images below shows the room I’m working in to solve this puzzle once and for all.

More is forthcoming and it will “build upon this article” already published:

That article built upon this one:

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OK - so I have made it through all the initial shapes with “4 in row”, and now I’m moving on to the shapes that formed from the 2nd shape (so to speak) made when studying shapes made with five (5)!

Shape 5 is below - it starts with “3 in a row with a 4th off to the side on the ‘bottom’ and a 5th off to the same side on the ‘top’ ”. It gives 7 new shapes with six (6) hexagons.

The above is was mostly “cut & paste” from what I already did, but nothing wrong with that is there…

Here is Shape #6:

Shape 6 is below - it starts with “3 in a row with a 4th off to the side on the ‘bottom’ and a 5th skipping a space above the 4th”. It gives 9 new shapes with six (6) hexagons.

So, there were 25 came from the first four shapes and now 7+9=16 from the next two giving what - 41 total.

Fore I go to bed on the night of 122123 - solstice happened today rumor I heard and the time where I reside just now tis 1047.

~~~~~~~~~~~~~~~~~ tis 122223 949 ~~~~~~~~~~~~~~~~~~~~~

Shape 7 is below - it is “three then two” (as shown). It gives 8 new shapes with six (6) hexagons.

Shape 8 is below - “you see it don’t you?”. It gives 8 new shapes with six (6) hexagons.

OK - we are now “up to 57 total. I’m confident that is more than half of the way!

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Kind readers - please alert me if you see a redundant shape or any other “mistake”. I need assistance with this - I got a few more in me today probably, and then I will be entering “virgin territory” so to speak…..but not really. Still, kinds readers please assist if there is something you notice and want to call to my attention!

~~~~~ tis 122223 1737 - here are Shapes 9 and 10 - edits may be necessary ~~~~~

The year ain’t over, but I’m going to be true to my word today and here are shapes #9 and #10 with no text….

(That is “minus 1” - now I got check closer….I’m guessing I missed one elsewhere - but I might be biased - tis 122823 - I’m in the midst of double-checking)

Another edit on 5224 - I found one been eliminated it was a mistake of a mistake - but something tells me - the final number is still 89, but now that some time has passed - I’m going to double check all this shape-imagination real and fixed - and come up with the final answer - after that - I plan on solving the puzzle when there are 7 shapes! I got some time to do this - and if nobody gives a shit about my shape imagination, then I reckon - what is the rush? If it turns out my “prime number” supposition is not correct, then I can live with that - cause I make mistakes - you do as well don’t ya? You don’t think you are infallible - do you? If you do, then hard lessons on the way for you direct. Not kidding around.

64! shapes so far I reckon.

More tomorrow if I got anything to say bout it - this is close if not 2/3rds of all the shapes there are going to be - how do I know you might wonder - check the links back to the original thread.

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I’m going to have to go back and seriously check for both redundant shapes or possibly any shape that might of been missed.

As evidenced by the fact that shapes 7-g and the one I had in the image (9-e is was) were and are “redundant” - no shape gets counted more than one time but differentiating the redundant ones demands a certain shape perspective that take practice and that is basically a big purpose of this puzzle. I think I am at 64 discrete shapes presently and there are about ?? (give or take) more shapes to be discovered, but the effort to do this is going to require a “new forum” and that is why as I proceed towards the solution fore the end of ‘23, I will continue the discussion on a completely new thread - a new post.

Well wishes to all and is there anything more fun then a puzzle harmless?

BK