Squares in tetta

  1. Latin Squares

    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.

    Latin square of order 3
    A B C
    B C A
    C A B
    Latin square of order 4
    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.

  2. Graeco-Latin Squares

    A Graeco-Latin 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 Graeco-Latin squares, the latter not.

    Graeco-Latin square of order 3
    A B C
    B C A
    C A B
    		+
    1 2 3
    3 1 2
    2 3 1
    		=
    A1 B2 C3
    B3 C1 A2
    C2 A3 B1
    Graeco-Latin square of order 4
    A B C D
    B A D C
    C D A B
    D C B A
    		+
    1 2 3 4
    3 4 1 2
    4 3 2 1
    2 1 4 3
    		=
    A1 B2 C3 D4
    B3 A4 D1 C2
    C4 D3 A2 B1
    D2 C1 B4 A3
    non-Graeco-Latin square of order 3
    A B C
    B C A
    C A B
    		+
    1 2 3
    2 3 1
    3 1 2
    		=
    A1 B2 C3
    B2 C3 A1
    C3 A1 B2
    non-Graeco-Latin square of order 4
    A B C D
    B A D C
    C D A B
    D C B A
    		+
    1 2 3 4
    2 3 4 1
    3 4 1 2
    4 1 2 3
    		=
    A1 B2 C3 D4
    B2 A3 D4 C1
    C3 D4 A1 B2
    D4 C1 B2 A3

    Rule 3 of tetta, both types of shapes - rectangles or triangles - are arranged as a Graeco-Latin square. The letters and numerals are replaced with colors.

  3. Numbers of Squares in the Three Rules of tetta

    Order n Total Permutations (n*n)! Rule 1 Column-Row 2*n!*n! Rule 2 Column/Row-Latin 4*L(n,n)*(n-1)!*n!*n! Rule 3 Graeco-Latin
    3 (3*3)!=362880 2*3!*3!=72 4*1*(3-1)!*3!*3!=288 72
    4 (4*4)!=20922789888000 2*4!*4!=1152 4*4*(4-1)!*4!*4!=55296 6912
    5 (5*5)!=15511210043330985984000000 2*5!*5!=28800 4*56*(5-1)!*5!*5!=77414400 ?
    6 (6*6)!=3.7199332678990121746799944815084e+41 2*6!*6!=1036800 4*9408*(6-1)!*6!*6!=2341011456000 0(*1)
    7 (7*7)!=6.082818640342675608722521633213e+62 2*7!*7!=50803200 4*16942080*(7-1)!*7!*7!=1239425105264640000 ?
    8 (8*8)!=1.2688693218588416410343338933516e+89 2*8!*8!=3251404800 4*535281401856*(8-1)!*8!*8!=17543398515000899272704000 ?
    9 (9*9)!=5.7971260207473679858797342315781e+120 2*9!*9!=263363788800 4*377597570964258816*(9-1)!*9!*9!=8.019287291701653098790924582912e+33 ?
    10 (10*10)!=9.3326215443944152681699238856267e+157 2*10!*10!=26336378880000 4*7580721483160132811489280*(10-1)!*10!*10!=1.4489707909649391634606365995965e+44 ?
    11 (11*11)!=8.0942985252734437396816228454494e+200 2*11!*11!=3186701844480000 4*5363937773277371298119673540771840*(11-1)!*11!*11!=1.2405611922440525795323213799888e+56 ?
    12 (12*12)!=5.5502938327393047895510546605504e+249 2*12!*12!=458885065605120000 4*1.62e+44*(12-1)!*12!*12!=5.93e+69(*2) ?
    13 (13*13)!=4.2690680090047052749392518888996e+304 2*13!*13!=77551576087265280000 4*2.51e+56*(13-1)!*13!*13!=1.86e+85(*2) ?
    14 (14*14)!=5.0801221108670467625027357853474e+365 2*14!*14!=15200108913103994880000 4*2.33e+70*(14-1)!*14!*14!=4.41e+102(*2) ?
    15 (15*15)!=1.2593608545945996091036028947807e+433 2*15!*15!=3420024505448398848000000 4*1.5e+86*(15-1)!*15!*15!=8.9e+121(*2) ?
    16 (16*16)!=8.5781777534284265411908227168123e+506 2*16!*16!=875526273394790105088000000 ? ?

    Remarks:

    n! = 1*2*3*...*n

    1. Graeco-Latin 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

  4. Patterns of Latin Squares

    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.

    the transformations of Latin squares of order 3
    e (123) (321) (12) (31) (23)
    1 2 3
    3 1 2
    2 3 1
    2 3 1
    1 2 3
    3 1 2
    3 1 2
    2 3 1
    1 2 3
    2 1 3
    3 2 1
    1 3 2
    3 2 1
    1 3 2
    2 1 3
    1 3 2
    2 1 3
    3 2 1

    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.

    the patterns of Latin squares of order 3
    123 132
    1 2 3
    2 3 1
    3 1 2
    1 2 3
    3 1 2
    2 3 1

    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.

    the patterns of Latin squares of order 4
    1234 1243 1324 1342 1423 1432
    1 2 3 4
    2 1 4 3
    3 4 1 2
    4 3 2 1
    1 2 3 4
    2 1 4 3
    4 3 2 1
    3 4 1 2
    1 2 3 4
    3 4 1 2
    2 1 4 3
    4 3 2 1
    1 2 3 4
    3 4 1 2
    4 3 2 1
    2 1 4 3
    1 2 3 4
    4 3 2 1
    2 1 4 3
    3 4 1 2
    1 2 3 4
    4 3 2 1
    3 4 1 2
    2 1 4 3
    1 2 3 4
    2 1 4 3
    3 4 2 1
    4 3 1 2
    1 2 3 4
    2 1 4 3
    4 3 1 2
    3 4 2 1
    1 2 3 4
    3 4 2 1
    2 1 4 3
    4 3 1 2
    1 2 3 4
    3 4 2 1
    4 3 1 2
    2 1 4 3
    1 2 3 4
    4 3 1 2
    2 1 4 3
    3 4 2 1
    1 2 3 4
    4 3 1 2
    3 4 2 1
    2 1 4 3
    1 2 3 4
    2 4 1 3
    3 1 4 2
    4 3 2 1
    1 2 3 4
    2 4 1 3
    4 3 2 1
    3 1 4 2
    1 2 3 4
    3 1 4 2
    2 4 1 3
    4 3 2 1
    1 2 3 4
    3 1 4 2
    4 3 2 1
    2 4 1 3
    1 2 3 4
    4 3 2 1
    2 4 1 3
    3 1 4 2
    1 2 3 4
    4 3 2 1
    3 1 4 2
    2 4 1 3
    1 2 3 4
    2 3 4 1
    3 4 1 2
    4 1 2 3
    1 2 3 4
    2 3 4 1
    4 1 2 3
    3 4 1 2
    1 2 3 4
    3 4 1 2
    2 3 4 1
    4 1 2 3
    1 2 3 4
    3 4 1 2
    4 1 2 3
    2 3 4 1
    1 2 3 4
    4 1 2 3
    2 3 4 1
    3 4 1 2
    1 2 3 4
    4 1 2 3
    3 4 1 2
    2 3 4 1

  5. Patterns of Graeco-Latin Squares

    Please observe the following transformations of Graeco-Latin 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 Graeco-Latin square of order 3, therefore there are 3!*3!=36 permutations in total; in other words, if the first row of the orginal Graeco-Latin 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 Graeco-Latin squares. Generally speaking, we can get n!*n! distinct Graeco-Latin squares through a pattern of Graeco-Latin squares of order n.

    the transformations of Graeco-Latin squares of order 3
    e (123) (321) (12) (31) (23)
    e
    A1 B2 C3
    B3 C1 A2
    C2 A3 B1
    A2 B3 C1
    B1 C2 A3
    C3 A1 B2
    A3 B1 C2
    B2 C3 A1
    C1 A2 B3
    A2 B1 C3
    B3 C2 A1
    C1 A3 B2
    A3 B2 C1
    B1 C3 A2
    C2 A1 B3
    A1 B3 C2
    B2 C1 A3
    C3 A2 B1
    (ABC)
    B1 C2 A3
    C3 A1 B2
    A2 B3 C1
    B2 C3 A1
    C1 A2 B3
    A3 B1 C2
    B3 C1 A2
    C2 A3 B1
    A1 B2 C3
    B2 C1 A3
    C3 A2 B1
    A1 B3 C2
    B3 C2 A1
    C1 A3 B2
    A2 B1 C3
    B1 C3 A2
    C2 A1 B3
    A3 B2 C1
    (CBA)
    C1 A2 B3
    A3 B1 C2
    B2 C3 A1
    C2 A3 B1
    A1 B2 C3
    B3 C1 A2
    C3 A1 B2
    A2 B3 C1
    B1 C2 A3
    C2 A1 B3
    A3 B2 C1
    B1 C3 A2
    C3 A2 B1
    A1 B3 C2
    B2 C1 A3
    C1 A3 B2
    A2 B1 C3
    B3 C2 A1
    (AB)
    B1 A2 C3
    A3 C1 B2
    C2 B3 A1
    B2 A3 C1
    A1 C2 B3
    C3 B1 A2
    B3 A1 C2
    A2 C3 B1
    C1 B2 A3
    B2 A1 C3
    A3 C2 B1
    C1 B3 A2
    B3 A2 C1
    A1 C3 B2
    C2 B1 A3
    B1 A3 C2
    A2 C1 B3
    C3 B2 A1
    (CA)
    C1 B2 A3
    B3 A1 C2
    A2 C3 B1
    C2 B3 A1
    B1 A2 C3
    A3 C1 B2
    C3 B1 A2
    B2 A3 C1
    A1 C2 B3
    C2 B1 A3
    B3 A2 C1
    A1 C3 B2
    C3 B2 A1
    B1 A3 C2
    A2 C1 B3
    C1 B3 A2
    B2 A1 C3
    A3 C2 B1
    (BC)
    A1 C2 B3
    C3 B1 A2
    B2 A3 C1
    A2 C3 B1
    C1 B2 A3
    B3 A1 C2
    A3 C1 B2
    C2 B3 A1
    B1 A2 C3
    A2 C1 B3
    C3 B2 A1
    B1 A3 C2
    A3 C2 B1
    C1 B3 A2
    B2 A1 C3
    A1 C3 B2
    C2 B1 A3
    B3 A2 C1

    The following 2 squares are the patterns of Graeco-Latin 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 Graeco-Latin squares, please derive them according to the above rules on your own.

    the patterns of Graeco-Latin squares of order 3
    123 132
    A1 B2 C3
    C2 A3 B1
    B3 C1 A2
    A1 B2 C3
    B3 C1 A2
    C2 A3 B1

    The following 12 squares are the patterns of Graeco-Latin 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 Graeco-Latin squares, please derive them according to the above rules on your own.

    the patterns of Graeco-Latin squares of order 4
    1234 1243 1324 1342 1423 1432
    A1 B4 C2 D3
    A1 B2 C3 D4
    C2 D1 A4 B3
    D3 C4 B1 A2
    B4 A3 D2 C1
    A1 B2 C3 D4
    C2 D1 A4 B3
    B4 A3 D2 C1
    D3 C4 B1 A2
    A1 B2 C3 D4
    D3 C4 B1 A2
    C2 D1 A4 B3
    B4 A3 D2 C1
    A1 B2 C3 D4
    D3 C4 B1 A2
    B4 A3 D2 C1
    C2 D1 A4 B3
    A1 B2 C3 D4
    B4 A3 D2 C1
    C2 D1 A4 B3
    D3 C4 B1 A2
    A1 B2 C3 D4
    B4 A3 D2 C1
    D3 C4 B1 A2
    C2 D1 A4 B3
    A1 B3 C4 D2
    A1 B2 C3 D4
    D2 C1 B4 A3
    B3 A4 D1 C2
    C4 D3 A2 B1
    A1 B2 C3 D4
    D2 C1 B4 A3
    C4 D3 A2 B1
    B3 A4 D1 C2
    A1 B2 C3 D4
    B3 A4 D1 C2
    D2 C1 B4 A3
    C4 D3 A2 B1
    A1 B2 C3 D4
    B3 A4 D1 C2
    C4 D3 A2 B1
    D2 C1 B4 A3
    A1 B2 C3 D4
    C4 D3 A2 B1
    D2 C1 B4 A3
    B3 A4 D1 C2
    A1 B2 C3 D4
    C4 D3 A2 B1
    B3 A4 D1 C2
    D2 C1 B4 A3

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