Continuing with the d[t]-d) analysis.
Pic related is reconfigured test case for c=145 showing values where:
DN = (d[t] - d) mod (n-1) == 0
AN = a mod n == 0
Not seeing anything yet.
See pics related for continued analysis of (e,1).
Examples are for c=65 and c=145.
Reminder for calculations we are comparing:
Calculation 1: (d[t]-d)/(n-1)
Calculation 2: a[t]/n
where d and n are the values from our entry record at (e,n). And d[t] and a[t] are from the (e,1) record being analyzed.
Only records where (n-1) and n are factors are shown.
Notes:
a) There are 2 sequences for each calculation.
b) One sequence increments by 1 and then is multiplied by a factor.
c) The other sequence increments by 2 and then is multiplied by a different factor.
d) Lines marked as "overlap" are where both calculations have (n-1) and n as factors.
Example:
c=145
the d[t] sequences are incremented by factors of 18 and 24.
the a[t] sequences are incremented by factors of 100 and 44.
No clue what to do with this information.
Also possible that for a/n, factor for sequence 1 comes from record (e,1,t+n) and factor for sequence 2 comes from record (e,1, n-t+1).
Yes. Very possible it adds up to d^2. Was just my initial observation.
Possible formula for the underlying factors to determine where a[t] % n == 0.
Pic related.
factor 1: 4*x
factor 2: (x-1)^2
Not sure if relevant.
Making a little progress.
Pics show various examples of the different factors for (d[t]-d) and a[t] formulas.
The formula (x^2 - f) can be used to find a valid factor of (d[t] - d). Sometimes it is the first valid factor below a record at (e,1, t+n).
How is this useful?
I believe we are searching for a formula for d.
For example, 901=17x53
Starting positions:
from c=901
(1,421,15) = {1:421:30:29:1:901}, f = 60
records at t = c.n + c.t
(1,17,436) = {1:17:23184:871:22313:24089}
(1,421,436) = {1:421:1772:871:901:3485}
Example formula that solves for the d value of (1,17,436) = 23184.
x^2-f = 29*29 - 60 = 781
(d)*(x^2 - f) - ( 2 * ( c - ( x^2 - f ) ) + 1 ) - 5
30(781) - (2(901 - 781) + 1) - 5
30*781 - (240+1) - 5
23430 - 241 - 5 = 23184
The 241 value is the next valid factor of d[t] - d after 781.
The 5 is just a value I added to the end because the numbers are really close. I was thinking of 2^2 + 1, or could be that one of the other formulas is wrong.
I understand the recursion, but it's not going to be the solution.
If you get a chance, take a look at some of my screenshots. There are a number of paths to matching records that I've identified, but none that are consistent for different test cases.
If I'm reading the QVC hint correctly, the solution can be calculated from the na record.
Found an interesting relationship between the c na record and the prime solution na record.
Pic is for c=145 example, but it applies everywhere.
Maybe this is obvious.
{1:1:72:11:61:85}
{1:1:32:7:25:41}
d = 72 - 32 = 40
x = 11 - 7 = 4
a = 61 - 25 = 36
b = 85 - 41 = 44
d-x = a
d+x = b
And if you look at the value of (x^2 - f) compared to the diff in d, the numbers are very close to a d[t] - d value in the list of valid factors of (d[t]-d) / (n-1).
Thanks VA.
Seems like the d[t] hint is exposing some real power behind the scenes in the grid. Wish we knew more.
VA - Excellent summary and understanding.
>is there supposed to be a (n-1)*a connection for d[t]?"
This is exactly what we are searching for. I still believe there is an elegant formula.
But notwithstanding the d[t] hint, I haven't found any combination of steps that is repeatable across different test cases.
Latest test case of c=145 attached.
Still looking at the differences between c at na, and prime at na.
Added tests to compare the differences in d, a, and b against differences in x.
(d-d)/(x-x)
(a-a)/(x-x)
(b-b)/(x-x)
Another humorous references to the boiling point of vinegar? Coincidence?
-x records seem to "line up better" than records in x. At least they give different target numbers to try and match to. See one of my sample pics.
You can create any corresponding -x record simply via an (e,n,t) formula.
for even e:
newT = 2 - t
for odd e:
newT = 1 - t
Same. My next step is to try and iron down the patterns for d[t] - d and a[t].
Attached are latest analysis of d[t]-d and a[t] for c=145, c=533 and c=6107. Just to give examples for various values of e.
These records are generated via a regular (e,1,t) function where t values begin from the original c at t=c.t + c.n, and then into negative t.
Included are differences for x and x/2, as well as those differences in a binary view where zeros are spaces, and 1's are #.
It looks like:
one of the sequences for a[t] is based on multiples of x.
one of the sequences for d[t]-d is based on multiples of (x+1).
There can be more than 2 interwoven sequences as the numbers get larger. See c=6107 as an example.
Not sure if this is adding any clarity.
>Are we stalled on finding a connection between c and p records? That jump between the two seemed like t-2 for the examples given.
VA - Unfortunately, the more I dig into the data, the further away a solution seems. So much to learn.
Thank you.
I've continued to look at the d[t] and a[t] differences and explored binary and factors of each.
Attached are latest tests for c=145, 533 and 6107.
I've added columns for t diff, t diff in binary, and t diff factors.
The factors are interesting, as they sometimes match the original x, x+1, x-1, d, d+1, or d-1 values indicated as [x], [x+], [x-], [d], [d+], [d-] respectively. I've also highlighted perfect square factors. Example is 4^(2) which indicates a 2x2 square.
Really interesting is the c=6107 example where a factor of 197 exists as a t diff factor in the d[t] - d analysis, and a factor of 31 exists as a t diff factor in the a[t] analysis.
I compared records at (e,1), (e,a), (e,b), (e,n), and (e,c) specifically because they share e, x and 2na values.
Figured that if we reduced the movement in variables, the solution would present itself. But I haven't been able to come up with a formula for d changes that accurately describes more than a few test cases.
I'm also leaning towards the offset being a difference in t or x. And will spend a bit more time on the (p * i + t, p * i + 1 - t) formulas you mentioned above.
Perhaps there is also something to be learned from looking at the difference between (-e,0) and the prime solution.
Just ran a quick test to confirm your records.
Here is some sample output for a=5, b=23. Also notice the -5 change in na.
(15,4,3) = {15:4:10:5:5:23} = 115; f=6; (x+n)=9; na=20; f+c=121; -x=-13
(-6,3,4) = {-6:3:11:6:5:23} = 115; f=29; (x+n)=9; na=15; f+c=144; -x=-12
(-29,2,4) = {-29:2:12:7:5:23} = 115; f=54; (x+n)=9; na=10; f+c=169; -x=-11
(-54,1,5) = {-54:1:13:8:5:23} = 115; f=81; (x+n)=9; na=5; f+c=196; -x=-10
(-81,0,5) = {-81:0:14:9:5:23} = 115; f=110; (x+n)=9; na=0; f+c=225; -x=-9
>There's only meant to be one value of negative e too, for a given static a and b.
This doesn't appear to correct. There is only 1 valid at n=0.
Thanks. That -x calculation is used to jump into the -x space. It's calculated as -(x + 2 * n); I'll remove as it's confusing.
Very interesting. Thank you for sharing this.
I've been able to reproduce the movement down to n=0.
Have you taken a look in the opposite direction?
newE = e + (2*d - 1)
newD = d - 1
newX = x -1
These entries literally go on for ever.
If I am following you correctly, your new e formula enables a jump directly to the corresponding n=0 record using values of e, n and d.
But how is this a solution when the a and b values are still unknown?
What am I missing?
Looked a bit further into (f,1) records, and analyzing the d[t] and a[t] differences.
See pics related for c=145, showing diffs at (e,1), (f,1) and (0,1).
In (f,1), there are more occurrences of the 5 factor in d[t] - d. But unfortunately, that pattern doesn't extend to other test cases. For c=6107, no records appear.
Similarly in (0,1) we get some matches at d[t]-d, but none at a[t].