Doing: lifting myself to hell and back and jump rope.
Building: Making maze generators in minetest (before moving them to minecraft)
And trying to figure out if I can find a function that takes two finite consecutive sequences with length N and M of natural numbers, which start by 1 (E.g [1,2,3], but not [1,2,4,5]) and give back a sequence which contain all the numbers starting from 1 up to the NxM, but not necessarily sorted without using the size of the sequences. So I want to number the cross product of the two sequences.
This crap is related to some programming problem I hit.
t = {1,2,3...} -- <- can be lenght
s = {1,2,3,4...} <- can be any length
xs = {}
for i,h in ipairs(t) do
for j,k in ipairs(s) do
q = f(i,j) -- <- I want to know if f is possible
to write
xs[q] = h * k
end
end
Well the answer is I think no, but I found some functions that work up to a certain number or that work within certain bounds, so how far we can stretch that? And of course it works for the whole set of natural numbers.
Pretty sure the answer is no as a math problem where f is pure. If f returns the same answer each time, the square ending at N,N contains [1, N * N]. Then the rectangle ending at N,N+1 must contain (N * N, N * (N+1)] in the nonoverlapping strip. Ditto for N+1,N. But then for N+1,N+1 all other locations must contain low numbers and there's only a single corner cell left which can contain the rest of the missing numbers between (N(N+1), (N+1)(N+1)]. Unless I made a mistake I don't think any function would work here for N > 1. What were the examples you found?
OTOH as a programming problem you can just cheat and store state somewhere to count the row width. This satisfies your interface requirements (python):
lastJ = None
def f(i, j):
global lastJ
if i != 0:
return i * (lastJ + 1) + j
else:
lastJ = j
return j
N = 3
M = 5
seen = set()
for i in range(N):
for j in range(M):
q = f(i, j)
seen.add(q)
assert sorted(seen) == list(range(N * M))
I am on work, so can't access my home laptop. But this problem is related to pairing functions (but those work for the whole domain of the naturals). It means you need to find a function that walks the numbers in a nice way, like seen here: http://szudzik.com/ElegantPairing.pdf
I think it isn't possible to find a function that works for arbitrary sequences. I know there is a function that works if the sets have the same size (n is size), it is trivial why those work. They have the form n^0 a + n^1 b, where a,b are in [1..n]:
f :: Integer -> (Integer,Integer) -> Integer
f n (i,j) = i + (j - 1)*n
-- For example:
f 3 <$> ((,) <$> [1..3] <*> [1..3])
[1,4,7,2,5,8,3,6,9]
Now I want to find functions like (n,m is size, n /= m):
f n m (i,j) = ..
And functions where n maybe chosen but m is between certain bounds. I have found a couple of those by mutilating pairing functions.
I know I can use a counter, but I don't like cheating :p
I need to draw your reasoning on a paper to see it or take some time for it. Little bit busy, working from home, having the kids and stuff :p
Doing: lifting myself to hell and back and jump rope.
Building: Making maze generators in minetest (before moving them to minecraft)
And trying to figure out if I can find a function that takes two finite consecutive sequences with length N and M of natural numbers, which start by 1 (E.g [1,2,3], but not [1,2,4,5]) and give back a sequence which contain all the numbers starting from 1 up to the NxM, but not necessarily sorted without using the size of the sequences. So I want to number the cross product of the two sequences.
This crap is related to some programming problem I hit.
Well the answer is I think no, but I found some functions that work up to a certain number or that work within certain bounds, so how far we can stretch that? And of course it works for the whole set of natural numbers.