Abstract: Sudoku is a game with incremental intelligence and is popularizing in many countries of Asia, Europe, and America. Its most common pattern is to fill the digits 1 through 9 into a square with 9 rows by 9 columns and subdivided 9 boxes, so that each digit appears once, and only once, in every row, column, and box. In the present paper, we investigated the basic properties of Sudoku in the general sense, and then make it a new design in field experiment.
A necessary condition for constructing a Sudoku square is k = pq and p, q ≥ 2, where k = number of rows, columns, and boxes in a k × k square, p = number of box-rows (the row consists of boxes) and q = number of box-columns (the column con-sists of boxes). Therefore, only non-prime number k can construct a Sudoku square and the prime number k can not. Table 1 lists 15 Sudoku squares in the case of k ≤ 20.
To design a Sudoku square, 4 steps are required as follows: (1) To write a k×k Sudoku square with restricted randomization , the procedure is to draw the random numbers of p or q sets each contains every number from 1 through k with no repeat, and then write down cyclically the first set numbers according to the order they appear into the first box-row or box-column, the second one into the second box-row or box-column, and so on and so forth. Note that any number must be shifted to the end of the series of random numbers if it has appeared in the column or row. (2) To randomize the order of box-rows and rows within each box-row. (3) To randomize the order of box-columns and columns within each box-column. (4) To assign randomly the k numbers to k treatments or k levels of a factor. Above mention indicates that a Sudoku square can layout k treatments with k replications and control 3-way (box, row, and column) soil-environment variations.
The linear model of data from a Sudoku square design is

where Y(ij)lm is an observed value of the plot in the lth row and mth column, subjected to the ith treatment and jth box; µ is the
overall mean; τi, βj, ρl, and γm are the main effects to the ith treatment, jth box, lth row, and mth column, respectively, and they
may be fixed or randomized; ε(ij)lm is random error and ε(ij)lm ~ N (0, ). Sudoku square can remove three sources of variation
from experiment error. Accordingly, the mean of treatment should be more precise with smaller error than that in Latin square.
Sudoku square design may be applied to multifactor experiments. The basic method is that the rows and/or columns may be substituted by experimental factors each consists of k levels, and hence the components ρl and γm in the model become main ef-fects of the factors. In such design, many effects and interactions have been confounded each other, but the main effects are or-thogonal and hence the estimate of one main effect is not influenced by the other main effects.

Key words: Sudoku square, Design and analysis, Field experiment