- Text
- Latin,
- Philatelic,
- Figure,
- Sudoku,
- Squares,
- Stamps,
- Pakistan,
- Puzzle,
- Square,
- Found,
- Comments,
- Pakjs.com

Some Comments on Philatelic Latin Squares from Pakistan

440 **on** **Philatelic** **Latin** **Squares** **from** **Pakistan** The Knut Vik designs KV1 and KV2 are also A-efficient 10 and pandiag**on**al, in that all diag**on**als (the two main diag**on**als and all broken diag**on**als) c**on**tain each of the five treatments precisely **on**ce. Moreover no two diag**on**ally-adjacent cells have the same treatment and, as noted by Bailey, Kunert & Martin [6], KV1 and KV2 are the **on**ly gerechte designs with this property. As Bailey, Camer**on** & C**on**nelly [4] point out, in 1956 [10] Walter- Ulrich Behrens 11 (1902–1962) introduced a specializati**on** of **Latin** squares which he called “gerechte” (Figure 4.2). Behrens [10, Tabelle 5b, p. 178] refers to the **Latin** square GD2 as a “diag**on**al layout” (German: Diag**on**alanordnung)—all the elements **on** the main backwards diag**on**al are equal (to 1), and [10, Tabelle 5a, p. 178] calls KV1 a “knight’s move layout” (German: Rösselsprunganordnung) and KV2 [10, Tabelle 8, p. 182] a “gerechte **Latin** square” (German: gerechte lateinische Quadrate). FIGURE 4.2: Gerechte **Latin** squares GD1 (left panel) and GD2 (right panel). In a k × k “gerechte design” the k × k grid is partiti**on**ed into k “regi**on**s” S1,S2,...,Sk, say, each c**on**taining k cells of the grid; we are required to place the symbols 1,2,...,k into the cells of the grid in such a way that each symbol occurs **on**ce in each row, **on**ce in each column, and **on**ce in each regi**on**. The row and column c**on**straints say that the soluti**on** is a **Latin** square, and the last c**on**straint restricts the possible **Latin** squares. Gerechte designs originated in the statistical design of agricultural experiments, where they ensure that treatments are fairly exposed to localized variati**on**s in the field c**on**taining the experimental plots. The Knut Vik designs KV1 and KV2 are gerechte designs of types GD1 and GD2, respectively. Bailey, Kunert & Martin [5, (1990)] point out that 10 The A-criteri**on** in the analysis of variance with a **Latin** square design is equivalent to minimizing: (i) the average variance of treatment-effect estimates, (ii) the average variance of an estimated difference in the treatmenteffect estimates, and (iii) the expected value of the treatment sum of squares when there are no treatment differences [55, p. 251]. 11 The well-known Behrens–Fisher problem of comparing means of two populati**on**s with possibly unequal variances was first c**on**sidered by this Walter-Ulrich Behrens [8] and Sir R**on**ald Aylmer Fisher (1890–1962) [39,40]. For recent results **on** the Behrens–Fisher problem see Ruben [69].

Ka Lok Chu et al. 441 Gerechte designs are row and column designs which have an additi**on**al blocking structure formed by spatially compact regi**on**s. They were popularized in Germany in 1956 by Behrens [9, 10]. We use the German word “gerechte” for these designs, as there is no simple word in English to express the idea of “fair-allocati**on**”, and because the designs originated in Germany, and do not appear to have spread **from** there. Behrens’s idea was to have a class of designs that were efficient, as good systematic designs are, but that allow some randomizati**on**. Soluti**on**s to 9 ×9 Sudoku puzzles are examples of gerechte **Latin** squares, where k = 9 and the regi**on**s S1,S2,...,S9 are the nine c**on**tiguous 3 × 3 regi**on**s (submatrices or boxes). Colbourn, Dinitz & Wanless [22] define a completed 9 × 9 Sudoku puzzle as a (3,3)- Sudoku **Latin** square. While Sudoku puzzles have become popular **on**ly very recently, Sudoku’s French ancestors **from** the late 19th century are discussed by Boyer [18, 19]; see also Bailey, Camer**on** & C**on**nelly [4] for an excellent survey of the c**on**necti**on**s between **Latin** squares and Sudoku, gerechte designs, resoluti**on**s, affine space, spreads, reguli, and Hamming codes. There are many books **on** Sudoku, at many levels. We particularly like the book How to Solve Sudoku, by Robin Wils**on** [87]. Most books **on** Sudoku c**on**centrate **on** the 9 × 9 puzzle, but Absolute Sudoku [91] has 4 × 4 and 6 × 6, as well as 9 × 9 puzzles. Sudoku Variants [23] has many different shapes of Sudoku puzzles including 15 puzzles of size 6 × 6 with various different “irregular” blocking patterns. We found very few books that include the smaller-size Sudoku puzzles. Thinkfun Thinkfun - Sudoku5X5 - Sudoku5X5 FIGURE 4.3: 5 × 5 Sudoku puzzle proposed by Maack [54] (left panel) and ThinkFun [79] (centre and right panels). Very little seems to have been written about 5 × 5 Sudoku puzzles: the **on**ly websites that we have found with such puzzles are by Maack [54] and ThinkFun [79]. The Sudoku- Download website created by A. Maack [54] c**on**tains Sudoku puzzles and soluti**on**s of For Retailers | Customer Service | Store Locator | Privacy Policy Â© 2006 ThinkFun, Inc. All Rights Reserved 17/06/09 17/06/09 11:03 11:03 AM AM

- Page 1 and 2: Pak. J. Statist. 2009, Vol. 25(4),
- Page 3 and 4: Ka Lok Chu et al. 429 design: a rev
- Page 5 and 6: Ka Lok Chu et al. 431 Depicted on t
- Page 7 and 8: Ka Lok Chu et al. 433 Pakistan issu
- Page 9 and 10: Ka Lok Chu et al. 435 We have found
- Page 11 and 12: Ka Lok Chu et al. 437 volcanic grou
- Page 13: Ka Lok Chu et al. 439 The Latin squ
- Page 17 and 18: Ka Lok Chu et al. 443 5 TWO MORE 5
- Page 19 and 20: Ka Lok Chu et al. 445 1 2 3 4 5 6 7
- Page 21 and 22: Ka Lok Chu et al. 447 6 PHILATELIC
- Page 23 and 24: Ka Lok Chu et al. 449 The puzzle S1
- Page 25 and 26: Ka Lok Chu et al. 451 6.2 Philateli
- Page 27 and 28: Ka Lok Chu et al. 453 where the 4
- Page 29 and 30: Hong Kong transport: 6x3 splits int
- Page 31 and 32: Ka Lok Chu et al. 457 6.4 Philateli
- Page 33 and 34: Ka Lok Chu et al. 459 1 2 3 4 5 6 7
- Page 35 and 36: y, (7) iris, (8) dahlia, (9) magnol
- Page 37 and 38: Ka Lok Chu et al. 463 6.6 Philateli
- Page 39 and 40: Ka Lok Chu et al. 465 7 ACKNOWLEDGE
- Page 41 and 42: Ka Lok Chu et al. 467 [27] Mikhail
- Page 43 and 44: Ka Lok Chu et al. 469 [57] P. J. Ow
- Page 45: Ka Lok Chu et al. 471 [92] Fuzhen Z