An algebraic reformulation: Given the table of two ternary numbers of order 1 with two figures:
21
12
We do the following operation: We replace in one column each figure with the two other ones, in the two different ways to make the next tables of order 1+1=2. We get then two tables with four ternary numbers of order 2 with 3 figures.
Operation on the 1st column:
011
101
022
202
Operation on the 2nd column:
202
220
101
110
We can continue for order n, and get then: n! tables with 2^n different ternary numbers each with n+1 figures. Proving that each table of some order has always at least two ternary numbers in common with any other table of the same order is one more generalization of the FCT. Here the two tables have 101 and 202 in common. Add a 0 to the right of every number in every table to see the relation with the illustration below.
Illustration of the operations above:
