A square grid of n symbols arranged in n rows and columns, such that each symbol appears once in each row and each column, is called a Latin square of order n. Such squares are of interest in encoding messages, in designing tournaments, and also as puzzles like Sudoku.

When adjacent elements in a row or a column are considered as ordered pairs, and if each such pair occurs only once, then the Latin square is called complete. It is well known that complete Latin squares exist for every even order. So far, for odd orders, existence has been proven only in certain composite cases. Beyond that, not much is known, and this paper extends this field with new results.

If the elements of a group can be arranged into a sequence such that product terms of the consecutive elements taken with the inverse of an element operated by the next one are all unique, then such an arrangement will be called a directed terrace, those product terms are called a sequencing, and the graph is called sequenceable. If a sequenceable group of order n exists, then a complete Latin square of order n can be constructed.

The main contribution in this paper is a construction of sequencing for groups whose order is thrice that of an odd powerful number. A number is called powerful if for any prime number that divides it, the square of that prime number also is a divisor. The constructions provided here also can provide sequencing for groups where the order is thrice a series of prime factors where each factor is congruent to 1 modulo 6.

The paper is well written and mathematically rigorous. After reviewing the basics quickly, the author takes the readers through a delightful ride of lemmas and theorems until the main result is proven. Though organized into sections based on the various cases under consideration, the material has continuity and needs to be followed in serial order.