The smallest possible tetramagic square
What is the smallest tetramagic (4multimagic) square possible? We don't know, we only know that its order is perhaps 24, or between 27 and 243.
The smallest known tetramagic square is order243, found in February 2004 by Pan Fengchu.
We will prove here that tetramagic squares of orders ≤ 23, or of orders 25 and 26, cannot exist.
11thorder tetramagic square? Or smaller order?
Since there is no 11th or lower order trimagic squares, an 11thorder or lower tetramagic square cannot exist.
Moreover, even a partial construction of an 11thorder tetramagic square is impossible, since there is no 11thorder tetramagic series.
12thorder tetramagic square?
There are exactly 106 series, so that could be enough to build a 12thorder tetramagic square, since 26 series (12 rows + 12 columns + 2 diagonals) "well chosen" could be sufficient. But they are not numerous enough. For example, the number 1 should be present in the square, and among the 106 series, there are only 5 series using 1:
Among these 5 series, the square needs two of them, one for the row + one for the column, intersecting on the number 1, no other number should be in common. This is impossible, any possible couple of these series has other numbers in common:

S1 
S2 
S3 
S4 
S5 
S1 
X 




S2 
44, 112, 120 
X 



S3 
112 
112, 125 
X 


S4 
50, 96 
135 
29, 54, 110 
X 

S5 
50, 112 
69, 112 
30, 110, 112 
38, 50, 110 
X 
So, a 12thorder tetramagic square cannot exist.
13thorder tetramagic square?
The magic sum S4 is 2152397897 = 9 mod 16. Because (2x+1)^4 = 1 mod 16 and (2x)^4 = 0 mod 16, each series has 9 odd numbers. So the 13 rows of the square would have 13x9 = 117 odd integers, even though in a 13thorder square we must place only 85 odd integers.
So, a 13thorder tetramagic square cannot exist.
14th or 15thorder tetramagic square?
There is no tetramagic series of such orders.
So, a 14th or 15thorder tetramagic square cannot exist.
16thorder tetramagic square?
There are 235275 series, and we may think that they could be sufficient, but we will prove that they can't be organized in a tetramagic square using modulo 9 reasoning. The magic sums of a 16thorder tetramagic square (S1 is not needed in the proof) are:
n 
n^{2} mod 9 
n^{3} mod 9 
n^{4} mod 9 
0 
0 
0 
0 
1 
1 
1 
1 
2 
4 
8 
7 
3 
0 
0 
0 
4 
7 
1 
4 
5 
7 
8 
4 
6 
0 
0 
0 
7 
4 
1 
7 
8 
1 
8 
1 
In a tetramagic series of 16 integers, calling a_{i} the number of its integers which are equal to i mod 9, and using the above table, we have this system of equations:
Analyzing all the possible solutions of this system, they are always:
Moving from mod 9 to mod 3, this also means that each tetramagic series has its 16 integers always split that way:
With such series, it is obviously impossible to get 16 rows using all integers from 1 to 256.
So, a 16thorder tetramagic square cannot exist.
17th, 18th or 19thorder tetramagic square?
On the 17th and 19thorders, Michael Quist sent me these short proofs in May 2008 using (2x+1)^4 = 1 mod 16 and (2x)^4 = 0 mod 16:
Though there may be tetramagic series of order 17, there cannot be a tetramagic square. Since S(4,17)%16 = 1, each tetramagic series of order 17 has either 1 or 17 odd elements. In 1...17^2 there are (17^21)/2 = 145 odd numbers to account for. This can be done only by having exactly 8 entirely odd rows, and 9 rows with a single odd element each. The same reasoning applies to the columns: there must be 8 entirely odd columns, and 9 columns with only a single odd entry. But since there are 8 entirely odd rows, each column must have at least 8 odd elements; there can't be any with just one odd entry.
So, a 17th, 18th or 19thorder tetramagic square cannot exist.
20th, 21st, 22nd or 23rdorder tetramagic square?
In FebruaryMarch 2013, Lee Morgenstern proved that 20th, 21st and 23rdorder tetramagic squares cannot exist. See his proofs (also including other proofs of orders ≥ 12)
24thorder tetramagic square?
May perhaps exist! Who will try?
25th or 26thorder tetramagic square?
In April 2013, Lee Morgenstern proved that 25thorder tetramagic squares cannot exist. See his proof.
27thorder tetramagic square?
May perhaps exist! Who will try?
Return to the home page http://www.multimagie.com