Estimates of the number of magic squares, cubes, ... (hypercubes)
Number of normal magic squares
Estimates using magic series
N(m) is the number of magic squares of order m.
For orders m ≥ 18 we may assume N(m) > 3 · E(m) and N(m) < C~(m) · E(m). (See diagram on the C-factor page.)
Probably the correct value is nearer to the upper value.
For m ≤ 20 these estimates are less accurate than the approximations by Monte Carlo Backtracking.
For m > 100 the approximation formula of Henry Bottomley was used to determine the number of magic series.