作物学报ACTA AGRONOMICA SINICA2008,39y1489-1493 http://www.chinacrops.org/zwxb/ SSN 0496-3490:CODEN TSHPA9 E-mail:xbzw@chinajournal.net.cn D0L10.3724/SPJ.1006.2008.01489 田间试验的一种新设计一数独方 莫惠栋许如根 (江苏省作物速传和生理学重点实验室/扬州大学，江苏扬州225009) 摘要：数独是近年流行的一种益智游戏，其最常见模式是在一个9行×9列又再分成9区共81个小格的方中，填 入适当的数字，使每一行、每一列、每一区都含有数字19，不重复。通过一般化地研究数独方的构成、设计、数学 模型和统计分析方法，使之成为田间试验的一种新设计。这种设计能够安排重复k次的k个处理或一个试验因素的k 个水平，能在行、列、区3个方向控制土壤环境的变异性，其处理平均数之间的比较将比拉丁方设计更为精确。 关键词：数独方，设计与分析；田间试验 Sudoku Square-A New Design in Field Experiment MO Hui-Dong and XU Ru-Gen (Jiangsu Provincial Key Laboratory of Crop Gcnetics and Physiology/Yangzhou University,Yangzhou) Abstract:Sudoku is a game with incremental intelligence and is popularizing in many countries of Asia,Europe,and America Its most common pattem is to fill the digits I through9intoasquare with9 rows by9columns 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 po and p.g2 2.where k number of rows,columns,and boxes in a k x k square,p=number of box-rows (the row consists of boxes)and g=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 I lists 15 Sudoku squares in the case of 20. To design a Sudokusquare,4steps e required as follows:(1)To write akk Sudoku sq stricted randomizatio he p m numbers of p or 4 sets ea through ith no repea and the Ny appe st be shifted to the numbers if it ha (3)To rand nd o with n.④To the k n ents or k levels of a fact Above mention indicates that a Sudoku squar an layout and column)soil The linear model of data from a Sudoku squre desig is Y+ +B+p+y+,,m=l,2,…,k) whereY is bserv ed valu row and mth col n,subjected to the ith treatment and /th box,u is th mcan,.p以 m are th box,/th row, ctively,and the fron ngly,the d than tha of and he fects of the fad s.In such desig effects and i thogonal e the estimate o main effect is not infuenced by the other main 基金项日：国家高技术研究发展计863计划项目(206AA10 国家自然科学茶金项日( 家科技支撑计刻项日2006BAD02B0 ped接受日期20006.i 研究方向： 生物统计学和数量传学。E 作物学报 ACTA AGRONOMICA SINICA 2008, 34(9): 1489−1493 http://www.chinacrops.org/zwxb/ ISSN 0496-3490; CODEN TSHPA9 E-mail: xbzw@chinajournal.net.cn DOI: 10.3724/SP.J.1006.2008.01489 田间试验的一种新设计——数独方 莫惠栋 许如根 (江苏省作物遗传和生理学重点实验室 / 扬州大学, 江苏扬州 225009) 摘 要: 数独是近年流行的一种益智游戏, 其最常见模式是在一个 9 行×9 列又再分成 9 区共 81 个小格的方中, 填 入适当的数字, 使每一行、每一列、每一区都含有数字 1~9, 不重复。通过一般化地研究数独方的构成、设计、数学 模型和统计分析方法, 使之成为田间试验的一种新设计。这种设计能够安排重复 k 次的 k 个处理或一个试验因素的 k 个水平, 能在行、列、区 3 个方向控制土壤-环境的变异性, 其处理平均数之间的比较将比拉丁方设计更为精确。 关键词: 数独方; 设计与分析; 田间试验 Sudoku Square—A New Design in Field Experiment MO Hui-Dong and XU Ru-Gen (Jiangsu Provincial Key Laboratory of Crop Genetics and Physiology / Yangzhou University, Yangzhou 225009, Jiangsu, China) 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 Y i (ij)lm i j l m (ij)lm =μ ++ + τ β ργ ε + + ( , , , =1, 2, ... , ), j l m k 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, 2 σ e ). 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. 基金项目: 国家高技术研究发展计划(863 计划)项目(2006AA10Z1C3); 国家自然科学基金项目(30671298); 国家科技支撑计划项目(2006BAD02B04) 作者简介: 莫惠栋(1934–), 男, 浙江温岭人, 教授, 博士生导师, 研究方向: 生物统计学和数量遗传学。E-mail: mhd28993@yahoo.com.cn Received(收稿日期): 2008-04-02; Accepted(接受日期): 2008-06-14