Estimates of the number of magic squares, cubes, ... (hypercubes) |
Multimagic squares |
N(m) ≈ N_{~}(m) = C_{~}(m) · E(m) | with the approximation C_{~}(m) = 0.185 · (m - 1) |
Number N of bimagic squares | ||
Order | What we estimated 2005 | What we know 2014 |
7 | 4 · 10^{-4} | N = 0 |
8 | 2 · 10^{07} | N = 26,158,848 |
9 | 1 · 10^{22} | N >> 1 |
10 | 2 · 10^{40} | N >> 1 |
11 | 3 · 10^{65} | N >> 1 |
12 | 4 · 10^{96} | N >> 1 |
I assume: the smallest strictly pandiagonal bimagic squares are of order 15. |
Number N of trimagic squares | ||
Order | What we estimate | What we know |
11 | 6 · 10^{-34} | N = 0 |
12 | 1 · 10^{-7} | N >> 1 |
13 | 2 · 10^{-1} | ? |
What is wrong with the trimagic order-12 estimate? |
E_{s,t}(12) ≈ 2.89 · 10^{125} · p_{row}^{11} · p_{col}^{7} / 8 ≈ 2.4 · 10^{7} |
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