A querie about order?

[samrw3]

[samrw3]

[samrw3]

[samrw3]

[samrw3]

[samrw3]

[samrw3]

Sam - you are a genius!

For practical purposes the numbers 1 to 6 would be perfect, since I am so familiar with the coler/number associations I have made that I am almost synesthetic in respect to them now. A list therefore of all 720 variations featuring the first six integers would be perfect!

Can't promise the finished work will be very good, but I give my word you'll get the chance to judge!

(It's my first abstract work, though I've done some acceptable figurative pieces before - strictly amateur I hasten to add!)

I hope that helps - I had to post as 18 total posts [18 posts X 40 rows = 720]. 40 was the easiest size for me to grab in a snapshot and still make it legible.

I am no genius - Microsoft Excel did the heavy lifting. It was a little bit of a slog getting through the variations for 1 but once I had those variations worked out I just copied the range of numbers after the first column[thus NOT including cells in C1 where 1 is residing) and copied it down to the 2 range then did find/replace. Made sure I had all cells except column C1 [where now 2 is residing] selected and found all 2's [making sure column C1 is not included because those are now cells with 2's that I want to keep] and replaced those 2's with 1's. Then just kept repeating this similar process through the 6 range.

For those that have spreadsheet knowledge I created a "Master" factorial table for a factorial of 6. Then I created a "Replacement" sheet/tab where I have 1 then next cell whatever the end user wants there. So Peter wanted 1-6 so the "Replacement" sheet looked like this

1 1
2 2
3 3
4 4
5 5
6 6

Then I created an area to perform a VLOOKUP where information was VLOOKUP from "Master" and "Replacement" to automatically change to however the end user wants to define the first 1 through 6.

Thus if Peter wanted to change to SUNDAY again the "Replacement" sheet would look like

1 S
2 U
3 N
4 D
5 A
6 Y

Now the VLookup will automatically change the results to have 720 variations of SUNDAY.

Peter - now get cracking on that abstract art!!

[peter]

Sam!


Can I just ask Sam (no more work to do I promise - this is probably more out of interest than of significance in respect of my plans for the piece) - does the order in which the sequences are given in your table {which from a brief inspection seems to be in numerically increasing order from smallest to largest} .............gosh, how can I put this .......... reflect the way in which the program has come up with the list. I mean, has it's mega-logical brain just started with the rules that a) it must look at all six figure numbers between 1 and 666666 and b) select every number in which all the integers 1,2,3,4,5 and6 appear .........and then simply started with 1 and gone through them one by one in it's super-quick computer fashion.....

Or is there some kind of logical system other than just simply looking at each integer one by one, by which the list can be arrived at?

(As a matter of interest, I tried yesterday to sit down with the number 123456 and then, by rearranging the individual numbers (first by moving the 1 down the line, then by doing the same with the 2 etc) come up with a tentative list, but within a few lines of sequences I was not only hopelessly lost, but repetitions were creeping in continuously.)

Sam - I must express my gratitude again; I was all but ready to drop this project and but for your help would no doubt have had to so. Your effort in preparing this list has I'm thinking, not been insignificant and you have my word that at my end it's reception is by no means trivial.


Now, there is work to do...........

(Edit; interesting - just spotted that the last two sequences do not seem to be in numerically increasing order (of no significance to the painting where the use of the sequences will be random) which might indicate a system other than rote trawling through the numbers one by one in order to generate the list.)

[samrw3]

I at first (weakly) tried to create a formula approach. But after rolling it over in my mind a couple minutes I realized that would be too complicated.

So... I just worked out the variations of the sequencing starting with 1 and then copied the section of results to sequence starting with 2. The I did a find and replace in the spreadsheet so that all 2's AFTER the first column [C1] were found and replaced by 1. Thus no complicated formula. In most spreadsheet application a find/replace literally takes a few seconds [unless you have gigantic data field]

How did I create the first 1 range of sequence?

If you look I copied 1 down 120 times, 2 down 120 times, 3 down a 120 times, etc. So now we have 1 --120 times, 2--120 times, etc. So now I am confident that I have 720 rows [120 X 6 = 720].

So then I went back up to the 1 sequence and starting in C2 I copied 2 down 24 times. Why 24 times? We know NON 2's in the 1- 6 range are 3,4,5,6 so this is factorial of 4 or 4! The result of factorial of 4! = 24.

Then in column C2 I copied 3 down 24 time, 4 down 24 times, 5 down 24 times and 6 down 24 times . So now in C2 I have 24 sequences of 2's, 24 sequences of 3's, 24 sequences of 4's, 24 sequences of 5's and 24 sequences of 6's. So now I have 24 cells each of 5 numbers (2,3,4,5,6) or 24 X 5=120.

So then I went back up to the 1 sequence and starting in C3 I copied 3 down 6 times. Why 6 times? We know NON 1's and NON 2's in the 1- 6 range are 4,5,6 so this is factorial of 3 or 3! The result of factorial of 3! = 6.

Then in column C3 I copied 3 down 6 times, 4 down 6 times, 5 down 6 times and 6 down 6 times . [24 rows total]

Now I am row 25 which started the 1 3 sequence so I copy 2 down 6 times, 4 down 6 times, 5 down 6 times and 6 down 6 times. [24 rows total]

Now I am row 49 which starts the 1 4 sequence so I copy 2 down 6 times, 3 down 6 times, 5 down 6 times and 6 down 6 times. [24 rows total]

Now I am row 73 which starts the 1 5 sequence so I copy 2 down 6 times, 3 down 6 times, 4 down 6 times, and 6 down 6 times. [24 rows total]

Now I am in row 97 which starts the 1 6 sequence so I copy 2 down 6 times, 3 down 6 times, 4 down 6 times and 5 down 6 times. [24 rows total]

So now in I have 5 instances of 24 rows 5 X 24 =120.

Then I went back up top to first row and column C4, and performed 2 sequences of each number [Except not 1's which are in C1] This was probably the trickiest column so it would take me forever to type in words how I typed the data into the spreadsheet. But I will attempt to describe it at high level and hope it is reasonable description. Since I knew it had to be the same number twice I just would type a number then go downwards to the next cell and type the same number. Always making sure to look to the left so as not to duplicate the numbers in C1, C2 and C3. So as I am typing along:

1234
1234
1235
1235
Just kept going down in similar manner and paying attention to when the sequence in C1 and C2 changed.

I admit C5 and C6 were also sort of tricky - so just the easiest way to describe how I did it was just patience and being careful. It actually looks a lot more difficult then it was for me to perform the steps - I just don't know how to easily tell you that I made sure that I had no errors.

So once I was certain I had all the correct variations for rows 1 - 120 - everything starting with 1. I took Cells ROW1-ROW120 Columns C2-C6 [purposely EXCLUDING C1] and pasted the cells into ROWS 121-240 C2-C6[again purposely EXCLUDING C1], Then did Find and Replace. Found all 2's in this section and replaced with 1's. [Replaced 2's in this section because now 2 is in C1]

So...your next question (anticipating it!) is HOW did I make sure there was no errors or repetition. Welllllll I built a nested IF statement
Or in normal terms I created 7 columns of calculations. The first column had calculation if Cell Row1 C1 has 1 then 1X1, if 2 then 2X2 if 3 then 3X3 if 4 then 4X4 if 5 then 5X5 if 6 then 6X6. I then copied that formula to the next five columns so I have 6 columns of same formula. Then in the seventh column I have the six prior columns added together [for those that are familiar with spreadsheets - yes I realize there is a different way I could have done these calculations - just was trying to make some of my steps easier and easier to try to describe here].

Example Row 1 1X1 = 1; 2X2=4; 3X3=9; 4X4=16; 5X5=25, 6X6=36
1+4+9+16+25+36=91. If I did everything right in columns C1 through C6 then the answer HAS to be 91. So 91 becomes my error checker.

Example Row 2 1X1 =1; 2X2=4; 3X3=9; 4X4=16; 6X6=36, 5X5=25
1+4+9+16+36+25=91

I made sure each 720 rows had result of 91 through this formula

I know this sounds like a lot of work but actually it sounds worse then it turned out to be.

Trying to describe in words how I performed data manipulation is actually trickier then performing the actual steps - so hopefully made some degree of sense.

[peter]

I'm overwhelmed and aghast Sam; never did I expect so much to be given in pursuit of 'the list'. This goes so far beyond me that I don't even know if what you did was math, computing or pure ratiocination taken to the nth degree. Irrespective of this however, I am acutely aware that thin words of gratitude can in no way cover such a gift and will say no more. You have done a fine thing for a complete stranger.

[Vraith]

Well, if you really want to cheat, you can just go here and type in the info...

https://www.mathsisfun.com/combinatoric ... lator.html

[peter]

Thanks V. That will come in handy if I opt for an ordered approach. (I have it in my head that the final effect of the finished piece work will depend much upon the ordering, or not, of the sequences. Time will tell if I have it right, but I have the idea of the painting 'reading' in a diagonal sweep from top left to bottom right, almost like a page of text. I've made a start, but there is much 'prep' to do before I get to actually applying 'colour to canvas'!

[samrw3]

Well, if you really want to cheat, you can just go here and type in the info...

https://www.mathsisfun.com/combinatoric ... lator.html

Cool Vraith! I was looking for something similar before I tried my approach .

Peter - all I can do is assure you that my approach sounds much worse then it was. To answer your question I used a combination of ratiocination, knowledge of spreadsheets and knowledge of factorial math.

The problem is me attempting to describe the approach in words. If we were able to sit next to each other at the time I did it - it would have made tons more sense. Trying to describe it in words in a short easy to digest way - not so much.

But your welcome - I found it a challenge so then it became interesting. I am just glad you kept to 6 because I don't think I would have tackled it my approach with 7.

An interesting idea for you - if you want randomness you can always take either Vraiths website or my posts - print them out. cut each row as a single slice of paper so now you have 720 slices of paper and find a way to mix all those slices and now you have a random ordered 720 slices of paper.

[Ur Dead]

Damm sam!
thats allota spreadsheet data.

[peter]

I'd actually thought of doing that Sam - except for I don't have a printer! What I've actually done is copied the list by hand and then gone through it to place the small number of cases where it was not so, into ascending order (I've decided upon a numerically ascending order of colour sequences). I love that 123456 is the smallest number on the list and 654321 the largest! This last process turned out to be really easy - by tackling the columns vertically left to right it was really quick to pick up the patterns and then spot discrepancies where they occured. By taking the two right hand columns two rows at a time, you'd always have say 3,4 above 4,3. If the lower figure was below the larger then there was some adjustment to make. (When copying the list, after a few pages I got good at her being able to write the numbers out for myself and then just backchecking to the list to make sure I'd got it right; the final thee figures in any sequence was particularly easy since you only had six variations after which the fourth from the right had to go up). Thus have we now created the numerical template which I can translate onto the paper.

[samrw3]

Cool looking forward to seeing the outcome!

[peter]

Me to Sam - but it'll take time. I made a start on the first half dozen sequences yesterday. Though not difficult to apply (though harder than one might imagine) the process is time consuming. It's by intention not perfectly uniform - and even the process of introducing spontaneity by this method demands attention.

[Hashi Lebwohl]

Well, if you really want to cheat, you can just go here and type in the info...

https://www.mathsisfun.com/combinatoric ... lator.html

I second this. WolframAlpha is a little more difficult to directly enter combinations and hypergeometrics.

I use hypergeometric fairly often when figuring out thing like "if I have 4 copies of card x in my MTG deck, what is the probability of having 1 of them in my opening hand?" (it's just about 40%, by the way).

[peter]

Is that what some of the guys call playing ' magic' over here Hashi? I've seen them playing in our local library - apparently some of the individual cards are worth shed-loads of money! Looks like fun, but I don't understand quite how it works.

[Fist and Faith]

peter, last year I sold my collection for $8,000. I did not have any of those shed-load cards, called Power, having sold them several years earlier. But many had gone up to the $100-200 range since then, and I had many many in the $5 range.