
The square of Fig.1 has a "[4]nucleus" ([4]nc), namely Fig.2. This means: Fig.1 is the only 5x5 magic square, which has the 4 entries at the 4 given places described by Fig. 2. So Fig.1 is uniquely determined by Fig.2. (Proof by computer experiment, see here f.i.). 

Example Fig.3 is a minimal [8]nc. There is only one 5x5 magic square with these clues, but if any of the entries 7, 12, 5, 25, 20, 23, 18, or 3 is deleted, then there will be more than one solution. 



with the two [4]nc's 


there exist exactly 101 magic 5x5 squares with these entries, therefore it is a [4][101]nc. The file "nc.htm" contains in its second table "1101" selected [4][m]nc's for m = 1, 2, ..., 101 determining 5x5 magic squares. The third table "rows" presents these [4][m]nc's written in rows of length 25. 
Proof by computer experiment. If is sufficient to deal with the 220 reduced 4x4 magic squares with a≤5, a<d,f,g,j,k,m,p and d<m as well as b<c. 


Fig.9, found by W. Trump, is an [8]nc with the only solution Fig.10. (The entry 16 cannot be left out, because of the exchange 20 ↔ 21 and 16 ↔ 17.) 

Fig.11 is a minimal [15]nc, none of its 15 entries can be omitted. (Proof by computer experiment, see here.) 

Definition When the entries in a nxn grid are denoted from left above to right below by c(1), c(2), ..., c(n^{2}), then a fixed point is an entry with c(i) = i. Fig.12 is a 6x6 magic square with a (non minimal) [18]nc, consisting of fixed points only. 

There are exactly four 5x5 magic squares with 13 fixed points (also found by W. Trump), and there is no 5x5 magic square with more than 13 fixed points. 




4x4 magic squares cannot have more than eight fixed points. 