KöMaL 2022 C1716

In factorial representation, the place values of the digits are not the powers of a base: the $n$th place value is $n$ factorial. Thus the digit in the first place is to be multiplied by 1, the digit in the second place is multiplied by 2, that in the third place is multiplied by 6, and so on. For example, the number $3310_!$ in factorial representation corresponds to the number $3\cdot4!+3\cdot3!+1\cdot2!=92$ in decimal notation. (If there is a number of more than one decimal digits in a certain place then it is indicated by using brackets.) (It can be shown that the representation is unique, that is, every positive integer has a single factorial representation. See the Informatics problems I. 553. of the January issue.) It is observed that one third of $111_!$ is $11_!$, one third of $111 111_!$ is $22 011_!$, and one third of $111 111 111_!$ is $33 022 011_!$. Determine the factorial representation of one third of the number that consists of $3n$ digits of 1, also given in factorial representation.