Multimagic Series Existence and Impossibility

Click on a hyperlink for the impossibility proof.

See Multimagic Series Examples for proof of existence.

order	bimagic	trimagic	tetramagic	pentamagic	hexamagic	heptamagic	octamagic
3	impossible	impossible	impossible	impossible	impossible	impossible	impossible
4	exists	exists	impossible	impossible	impossible	impossible	impossible
5	exists	exists	impossible	impossible	impossible	impossible	impossible
6	exists	impossible	impossible	impossible	impossible	impossible	impossible
7	exists	impossible	impossible	impossible	impossible	impossible	impossible
8	exists	exists	impossible	impossible	impossible	impossible	impossible
9	exists	exists	impossible	impossible	impossible	impossible	impossible
10	exists	impossible	impossible	impossible	impossible	impossible	impossible
11	exists	exists	impossible	impossible	impossible	impossible	impossible
12	exists	exists	exists	exists	impossible	impossible	impossible
13	exists	exists	exists	exists	impossible	impossible	impossible
14	exists	impossible	impossible	impossible	impossible	impossible	impossible
15	exists	exists	impossible	impossible	impossible	impossible	impossible
16	exists	exists	exists	exists	impossible	impossible	impossible
17	exists	exists	impossible	impossible	impossible	impossible	impossible
18	exists	impossible	impossible	impossible	impossible	impossible	impossible
19	exists	exists	exists	impossible	impossible	impossible	impossible
20	exists	exists	exists	exists	impossible	impossible	impossible
21	exists	exists	exists	exists	impossible	impossible	impossible
22	exists	impossible	impossible	impossible	impossible	impossible	impossible
23	exists	exists	exists	impossible	impossible	impossible	impossible
24	exists	exists	exists	exists	impossible	impossible	impossible
25	exists	exists	exists	impossible	impossible	impossible	impossible
26	exists	impossible	impossible	impossible	impossible	impossible	impossible
27	exists	exists	exists	exists	unknown	unknown	impossible
28	exists	exists	exists	exists	impossible	impossible	impossible
29	exists	exists	exists	exists	impossible	impossible	impossible
30	exists	impossible	impossible	impossible	impossible	impossible	impossible
31	exists	exists	unknown	impossible	impossible	impossible	impossible
32	exists	exists	exists	exists	impossible	impossible	impossible
33	exists	exists	unknown	unknown	impossible	impossible	impossible
34	exists	impossible	impossible	impossible	impossible	impossible	impossible
35	exists	exists	unknown	unknown	impossible	impossible	impossible
36	exists	exists	unknown	unknown	impossible	impossible	impossible
37	exists	exists	unknown	unknown	impossible	impossible	impossible
38	exists	impossible	impossible	impossible	impossible	impossible	impossible
39	exists	exists	unknown	unknown	impossible	impossible	impossible
40	exists	exists	exists	exists	unknown	unknown	impossible
41	exists	exists	unknown	unknown	unknown	unknown	impossible
42	exists	impossible	impossible	impossible	impossible	impossible	impossible
43	exists	exists	unknown	unknown	impossible	impossible	impossible
44	exists	exists	unknown	unknown	impossible	impossible	impossible