The problem with large boards is not so much the computation of the result but its presentation as an image. By doing the output as a mosaic that can be stitched together and downscaled, I can get at least up to 32001 x 32001, here scaled down to 4000 x 4000.
I was able to push this to 24001 x 24001 squares/pixels. (This isn't the full resolution image, it's downscaled to 3000 x 3000.) I don't know how to do anything larger. Anyway, the irregular bendy radial stripes seem to have straightened themselves out and maybe they do go to infinity like that. Maybe there's a way to prove they do.
Still more of these bizarre things.
Another red and black "knights" image. Here both attack (4,0), (4,1), and (5,0) and their rotations into the other quadrants. The board is 4001 x 4001.
Falling asleep the other night I was thinking about derivatives.
I can think of two functions that are their own derivatives: f(x) = 0 and f(x) = ex.
There's a pair of functions function such that each is its second derivative is itself: f(x) = e-x and g(x) = -e-x.
There's a group of functions such that its fourth derivative is itself, f(x) = sin(x), g(x) = cos(x), h(x) = -sin(x), k(x) = -cos(x)
Are there any functions with other numbers of derivative steps before they become equal to themselves again? 3, 5, 8, etc.
In the light of day, I think I see that we could define an "n-self-derivative function" by just setting the derivatives
Rich Holmes
Stylusf(0) = 1
f'(0) = -1/2
f''(0) = -1/2
f'''(x) = f(x) etc
and that defines a unique Maclurin series.
But .. are there functions of this form that would arise naturally?
Update: Yes, one such function is exp(-x/2)*cos(x*sqrt(3)/2). see the response from @Chip_Unicorn: https://im-in.space/@Chip_Unicorn/116529978344298695
@[email protected] Math stack exchange has the answer: https://math.stackexchange.com/questions/1080016/function-whose-third-derivative-is-itself The answers are interesting!
I have made a symbol to represent 𝛕 (=2π), it is a scheme of how a circle opens and straightens. In the second figure more steps are added, the curve defined by the endpoints is a cochleoid. Finally the third figure shows a tau spider.
UPDATE: My program found a new personal best most delayed palindromic number!
The 25-digit number 1102320000659911613429286 takes 235 iterations before it becomes a 123-digit palindrome.
This beats a former world record set in 2003 by Jason Doucette involving 233 iterations (source: https://jasondoucette.com/worldrecords.html#Most), and is 58 iterations off from the current world record of 293.
#Math #LychrelNumber #MostDelayedPalindromicNumber #Mathematics #RecMath #RecreationalMathematics
((10^n)-1)*(2/3)-53, a formula resulting in rows of n number of sixes with the last two sixes replaced with the number 13, result in primes for n = 2, 3, 7, 12, 30, 97, 192, 265, 327, 417, 475, 574, 595, 699, 9563, and 9601. 
Update: It's ran for 10 days and hasn't found any other new primes below n = 37000. Stopping the program now!
#Math #Mathematics #PrimeNumbers #RecreationalMath #RecMath #RecreationalMathematics #Primes
Dani Laura (they/she/he)
greatquux
MOULE 