A Latin square of order n is a n*n square filled with n letters or numerals such that letters or numerals in each column or row are distinct. Becauce they are distinct, each letter or numeral occurs n times in the square. The following are Latin squares of order 3 and 4.
A  B  C 
B  C  A 
C  A  B 
1  3  2  4 
2  1  4  3 
3  4  1  2 
4  2  3  1 
Rule 2 of tetta, a type of shapes in cells  rectangles or triangles  are arranged as a Latin square. The letters or numerals are replaced with colors. Another type of shape are arranged as the same color in each column or row.
A GraecoLatin square of order n is combined with two Latin squares of order n. The combinations of letters and numerals of cells in the new square are distinct. The following are four squares of order 3 and 4 each of which is combined with two Latin squares. the former two are GraecoLatin squares, the latter not.

+ 

= 


+ 

= 


+ 

= 


+ 

= 

Rule 3 of tetta, both types of shapes  rectangles or triangles  are arranged as a GraecoLatin square. The letters and numerals are replaced with colors.
Order n  Total Permutations (n*n)!  Rule 1 ColumnRow 2*n!*n!  Rule 2 Column/RowLatin 4*L(n,n)*(n1)!*n!*n!  Rule 3 GraecoLatin 

3  (3*3)!=362880  2*3!*3!=72  4*1*(31)!*3!*3!=288  72 
4  (4*4)!=20922789888000  2*4!*4!=1152  4*4*(41)!*4!*4!=55296  6912 
5  (5*5)!=15511210043330985984000000  2*5!*5!=28800  4*56*(51)!*5!*5!=77414400  ? 
6  (6*6)!=3.7199332678990121746799944815084e+41  2*6!*6!=1036800  4*9408*(61)!*6!*6!=2341011456000  0(*1) 
7  (7*7)!=6.082818640342675608722521633213e+62  2*7!*7!=50803200  4*16942080*(71)!*7!*7!=1239425105264640000  ? 
8  (8*8)!=1.2688693218588416410343338933516e+89  2*8!*8!=3251404800  4*535281401856*(81)!*8!*8!=17543398515000899272704000  ? 
9  (9*9)!=5.7971260207473679858797342315781e+120  2*9!*9!=263363788800  4*377597570964258816*(91)!*9!*9!=8.019287291701653098790924582912e+33  ? 
10  (10*10)!=9.3326215443944152681699238856267e+157  2*10!*10!=26336378880000  4*7580721483160132811489280*(101)!*10!*10!=1.4489707909649391634606365995965e+44  ? 
11  (11*11)!=8.0942985252734437396816228454494e+200  2*11!*11!=3186701844480000  4*5363937773277371298119673540771840*(111)!*11!*11!=1.2405611922440525795323213799888e+56  ? 
12  (12*12)!=5.5502938327393047895510546605504e+249  2*12!*12!=458885065605120000  4*1.62e+44*(121)!*12!*12!=5.93e+69(*2)  ? 
13  (13*13)!=4.2690680090047052749392518888996e+304  2*13!*13!=77551576087265280000  4*2.51e+56*(131)!*13!*13!=1.86e+85(*2)  ? 
14  (14*14)!=5.0801221108670467625027357853474e+365  2*14!*14!=15200108913103994880000  4*2.33e+70*(141)!*14!*14!=4.41e+102(*2)  ? 
15  (15*15)!=1.2593608545945996091036028947807e+433  2*15!*15!=3420024505448398848000000  4*1.5e+86*(151)!*15!*15!=8.9e+121(*2)  ? 
16  (16*16)!=8.5781777534284265411908227168123e+506  2*16!*16!=875526273394790105088000000  ?  ? 
n! = 1*2*3*...*n
1. GraecoLatin squares of order 6 do not exist.
2. Estimated values.
? Not zero, but unknown.
Generally speaking, numbers of rule 2 > numbers of rule 3 > numbers of rule 1
Please observe the following transformations of Latin squares of order 3. The leftmost e of them means no transformation, ie the original square; (123) means 1→2→3→1, ie changing numerals in the original square; (321) means 3→2→1→3; (12) means 1↔2; (31) means 3↔1; (23) means 2↔3. Since there are 3 numerals in a Latin square of order 3, therefore there are 3!=6 permutations in total; in other words, if the first row of the orginal Latin square is fixed at 123, and then the 3 numerals are renewed and permutated, we can get 3!=6 distinct Latin squares. Generally speaking, we can get n! distinct Latin squares through a pattern of Latin squares of order n.
e  (123)  (321)  (12)  (31)  (23)  







The following 2 squares are the patterns of Latin squares of order 3. The first rows are both 123; 123 means the first column is 123; 132 means the first column is 132. Each pattern represents 3!=6 distinct Latin squares, please derive them according to the above rules on your own.
123  132  



The following 24 squares are the patterns of Latin squares of order 4. The first rows of squares are all 1234; 1234 means the first columns of squares are 1234, ie the leftmost original square; 1243 means the first columns of squares are 1243, ie interchanging the 3rd and 4th rows of the leftmost original square; 1324 means the first columns of squares are 1324, ie interchanging the 2nd and 3rd rows of the leftmost original square; and so on. There are 4 squares in each column, ie although the first columns of squares are fixed, but there are 4 distinct patterns. Each pattern represents 4!=24 distinct Latin squares, please derive them according to the above rules on your own.
1234  1243  1324  1342  1423  1432  




























Please observe the following transformations of GraecoLatin squares of order 3. See the above paragraph for the meanings of notations. The left top is the original square. Of the 6 squares in each column, the below 5 squares are derived by permutating the letters in the topmost square; Of the 6 squares in each row, the right 5 squares are derived by permutating the numerals in the leftmost square. Since there are 3 letters and 3 numerals in a GraecoLatin square of order 3, therefore there are 3!*3!=36 permutations in total; in other words, if the first row of the orginal GraecoLatin square is fixed at A1 B2 C3, and then the 3 letters and 3 numerals are renewed and permutated, we can get 3!*3!=36 distinct GraecoLatin squares. Generally speaking, we can get n!*n! distinct GraecoLatin squares through a pattern of GraecoLatin squares of order n.
e  (123)  (321)  (12)  (31)  (23)  

e 







(ABC) 







(CBA) 







(AB) 







(CA) 







(BC) 






The following 2 squares are the patterns of GraecoLatin squares of order 3. The first rows are both A1 B2 C3; 123 means the first column is 123; 132 means the first column is 132. Each pattern represents 3!*3!=6 distinct GraecoLatin squares, please derive them according to the above rules on your own.
123  132  



The following 12 squares are the patterns of GraecoLatin squares of order 4. The first rows of squares are all A1 B2 C3 D4; 1234 means the first columns of squares are 1234, ie the leftmost original square; 1243 means the first columns of squares are 1243, ie interchanging the 3rd and 4th rows of the leftmost original square; 1324 means the first columns of squares are 1324, ie interchanging the 2nd and 3rd rows of the leftmost original square; and so on. A1 B4 C2 D3 means the first columns are all composed of the 4 combinations (maybe permutated); A1 B3 C4 D2 means the first columns are all composed of the 4 combinations (maybe permutated). Each pattern represents 4!*4!=576 distinct GraecoLatin squares, please derive them according to the above rules on your own.
1234  1243  1324  1342  1423  1432  

A1 B4 C2 D3 







A1 B3 C4 D2 






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