Nuclei for Magic Squares
© H.B. Meyer         magic squares and cubes

Fig.1

25  1  2 14 23 18 12  3 10 22  7 20 15 17  6 11 13 24  8  9  4 19 21 16  5

 is a (classical) 5x5 magic square:
 - each of the integers 1, 2, ..., 25 appears
 - the 5 rows, the 5 columns and the 2 diagonals sum up to the same result (=65).

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.).

Fig.2        1  2 14        3

Definition A choice of k places provided with k entries from {1, 2, ..., n} in a nxn grid is called a [k]-nucleus ([k]-nc) (for the nxn grid), when there is a unique nxn magic square with the given k entries at the given k places.
A [k]-nc is called minimal if no entry can be left out without losing uniqueness of the magic square solution.

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.

Fig.3      7 12  5 25   20 23 18  3

For 5x5 magic squares [4]-nc's appear to be quite rare. The square

Fig.4

1  3 24 15 22 19 17 14 10  5 20 18 13  8  6 21 16 12  9  7  4 11  2 23 25

(the lexicographically first central symmetric 5x5 magic square) does not have a [4]-nc, moreover, it has no [5]-nc. Therefore, for any choice of 5 places from Fig.4 together with their associated entries, there exists another 5x5 magic square (different from Fig.4) with these entries at these places (proof by computer experiment). The 6 entries 1, 3, 10, 14, 18, and 23 in Fig.4, however, form a [6]-nc for Fig.4.

On the other hand, there are 5x5 magic squares with at least two different [4]-nc's. An example is:

Fig.5  4 17 24 15  5 10 14 16 18  7 23 12  1  9 20  6 19 11 21  8 22  3 13  2 25

with the two [4]-nc's

Fig.6  4     15  5   14            1

where the underlined entries 14 and 15 can be used alternatively.

Observation The image of a [k]-nc under any map, which maps every nxn magic square onto a nxn magic square again, is a [k]-nc, too. Any set of chosen k places in a nxn grid, which is mapped onto itself by any such map, different from identity, cannot produce a [k]-nc.

Proposition There is no 5x5 magic square with a [3]-nc. This means: for every choice of three places from a 5x5 grid and every choice of three integers from {1, 2, ..., 25} as entries, there exists either none or more than one 5x5 magic square with these entries at these places.
Proof by computer experiment. It is sufficient, to consider pairwise non equivalent sets of 3 places. Two sets of 3 places are equivalent, if they can be mapped onto each other by one of the 32 mappings for 5x5 magic squares. Furthermore it is possible to consider only sets of 3 places not invariant under any such map, different from identity. Alltogether 71 sets of 3 places have to be observed.

Problem Find a distribution of the numbers 1, 2, 3, and 4 in a 5x5 grid, producing a [4]-nc.

The situation of [k]-nc's seems to be similar to the situation of SUDOKU. The uniqueness of the solution derived from given entries is essential.

The file "nc.htm" (also as Excel file "nc.xls") contains in its first table "4-nc's" a collection of several [4]-nc's for 5x5 magic squares.

Generalisation A choice of k places provided with k entries from {1, 2, ..., n} in a nxn grid is called a [k]-[m]-nucleus ([k]-[m]-nc) (for the nxn grid), when there are exactly m magic nxn squares with the given k entries at the given k places. (a [k]-[1]-nc is a [k]-nc)

Example For

Fig.7    1       10      9          2

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 "1-101" 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.

4x4 magic squares:

Proposition There exists no 4x4 magic square with a [2]-nc. Exactly 88x32 = 2816 of the 7040 magic 4x4 squares possess a [3]-nc, the remaining 4224 ones have a [4]-nc each.
The squares with [3]-nc belong to the Dudeney types VI, VII, ..., XII, squares of Dudeney types I, II , III, IV, and V do not allow [3]-nc's.

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.8   a b c d e f g h i j k l m n o p

Problems
a) Find a distribution of the numbers 1, 2, and 3 in a 4x4 grid, producing a [3]-nc.
b) find a distribution of the numbers 1, 2, 3, and 4 in a 4x4 grid, producing a minimal [4]-nc.

In the table "88" of file "nc.htm" the 88 reduced 4x4 magic squares with a [3]-nc can be found in rowwise representation together with the entries of their chosen [3]-nc's and their Dudeney types.
The table "220" contains all 220 reduced 4x4 magic squares together with the entries of their [3]- resp. [4]-nc's.

6x6 magic squares To obtain [k]-nc's with small k or minimal [k]-nc's with large k for nxn grids, 6≤n, seems to be difficult (see challenges below).

Fig.9  1          4    2     12        3            5     16   25

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.10  1 29 31 18 28  4 26  2 30 14 12 27 20 24  3 32 15 17 21 22  5 34 13 16 10 25 23  7 35 11 33  9 19  6  8 36

Fig.11 is a minimal [14]-nc, none of its 14 entries can be omitted. (Proof by computer experiment, see here.)

Fig.11      7 15 23 34     18  8 33       26 12 29 30          6 24   25

fixed points

Definition When the entries in a nxn grid are denoted from left above to right below by c(1), c(2), ..., c(n2), 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.

Fig.12      1 33 35  4 32  6 27  8 19 34 11 12 24 14 15 16 17 25  5 20 21 22 13 30 23 26  3 28 29  2 31 10 16  7  9 36

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.

Fig.13      1 15 24  4 21 23  7  8 17 10 20 12 13 14  6 16  9 18 19  3  5 22  2 11 25

Fig.14      1 21  3 18 22 24  7 15  9 10 20 12 13 14  6 16 17 11 19  2  4  8 23  5 25

Fig.15      1 21 24  4 15 17  7  8 23 10 20 12 13 14  6 16  3 18 19  9 11 22  2  5 25

Fig.16     25  2 22 11  5  6 19  8  9 23 10 12 13 14 16  3 17 18  7 20 21 15  4 24  1

Fig.17 1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16

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

Challenges
a) Does every 5x5 magic square possess a [6]-nc ?
b) Can a 5x5 magic square be found with a minimal [k]-nc and 9≤k ?
c) What is the smallest number k, such that there exists no 6x6 magic square with a [k-1]-nc ?
d) Determine the largest number k, such that there is a 6x6 magic square with a minimal [k]-nc.
e) What is the smallest number k, such that every 6x6 magic square has a [k]-nc ?
f) How many fixed points can a nxn magic square have for 6≤n ?

Last update: 2018/October/20