Construct a bijective formula between N and NxNxN Construct

Solution

If a function is bijection f^-1 exists and it is also bijective.

So define

Suppose g is a bijection from N × N to N and let g^-1 be its inverse. Then we can define f in terms of g as follows.

f(x, y, z) = g(g(x, y), z) = n

f^-1(n) = (x, y, z) where g^-1(n) = (w, z) and g^-1(w) = (x, y)

Defining g as a bijection from N × N to N

We now have the much simpler problem of defining g.

g(x, y) = (x + y)(x + y + 1) / 2 + y = n

g^-1(n) = (x, y) where m = (2n)^1/2 and exactly one of the following two conditions hold.

x + y = m and y = n - m(m + 1) / 2

x + y = m - 1 and y = n - m(m - 1) / 2