2007-12-66

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Feature not have a 1, and then the 1 in row 2 must be in cell c. But then the colored area cannot contain a 1, impossible. (So, cell a has a 5.) Forcing Chains Consider propositions (i, j)d ‘cell (i, j) has value d’and(i, j)!d ‘cell (i, j) has a value different from d’. By observing the grid one finds implications among such propositions. There are at least three obvious types of such implications. Let us say that two cells are ‘adjacent’ (or, ‘see each other’) when they lie in the same row, column or box, so that they must contain different digits. This gives the first type: If (i, j) is adjacent to (k,l) then (i, j)d > (k,l)!d where > denotes implication. In case (i, j) is adjacent to (k,l) and (k,l) only has the two possibilities d and e, then (i, j)d > (k,l)e. This is the second type of implication. Finally, if some row, column or box has only two possible positions (i, j) and (k,l) for some digit d, then(i, j)!d > (k,l)d. This is the third type. Consider chains of implications. If (i, j)d >... >(i, j)!d then (i, j)!d. For example Uniqueness A properly formulated Sudoku puzzle has a unique solution. One can assume that a given puzzle actually is properly formulated, and use that in the reasoning, to exclude branches that would not lead to a unique solution. For example, a grid can be completed in at least two ways, violating the uniqueness assumption. That means that this must be avoided, so that At least one of the corners of the rectangle is 2. For example, Here (8,2)6 > (8,6)8 > (5,6)6 > (5,1)5 > (6,2)6 > (8,2)2 was used to conclude (8,2)2. For example Here (8,2)1 > (6,2)!1 > (6,9)1 > (4,7)!1 > (8,7)1 > (8,2)!1 was used to conclude (8,2)!1. For such chains that involve a single digit only, and where the implication types alternate between I and III, one often uses a simplified notation like 1 : (8,2) − (6,2) =(6,9) − (4,7) =(8,7) − (8,2) where − denotes that at most one is true and = that at least one is true. Finding useful chains may be nontrivial, and there are various techniques such as ‘colouring’ that help. Look at the colored rectangle. If (7,7)3, then (7,5)8 and (8,7)8sothat(8,5)3 and we have a forbidden rectangle with pattern 83–38. So, (7,7)!3, which means that we have an X- wing: digit 3 in columns 2,7 can only be in rows 8,9 and does not occur elsewhere in these rows. In particular (9,4)!3 so that (6,4)3, and (8,5)!3 so that (8,5)8. More generally: Theorem. Suppose one writes some (more than 0) candidate numbers in some of the initially open cells of a given Sudoku diagram, 0 or 2 in each cell, such that each value occurs 0 or 2 times in any row, column, or box. Then this Sudoku diagram has an even number of completions that agree with at least one of the candidates in each cell where candidates were 16 EMS Newsletter December 2007
Feature given. In particular, if the Sudoku diagram has a unique solution, then that unique solution differs from both candidates in at least one cell. Digit patterns and jigsaw puzzles A more global approach was described by . Solve a puzzle until no further progress is made. Then, for each of the nine digits, write down all possible solution patterns for that digit. One hopes to find not more than a few dozen patterns in all. Now the actual solution has one pattern for each digit, where these 9 patterns partition the grid. Regard each digit pattern (‘jigsaw piece’) as a boolean formula (‘this pattern occurs in the solution’). Write down the formulas that express that for each of the nine digits exactly one pattern occurs, and that overlapping pieces cannot both be true. Solve the resulting system of propositional formulas. This approach allows one to solve some otherwise unapproachable puzzles. new to oxford journals in 2007 INTERNATIONAL MATHEMATICS RESEARCH NOTICES www.imrn.oxfordjournals.org Bibliography [1] There is an enormous amount of literature on Sudoku on the web. This note is a condensed version of http://homepages.cwi. nl/~aeb/games/sudoku/. This article has previously appeared in Nieuw Archief voor Wiskunde (5) 7, no. 4, pp. 258–263, December 2006. We thank the editor for the permission to reproduce it in the Newsletter. Andries E. Brouwer [aeb@cwi.nl/aeb@win. tue.nl] is a professor of mathematics at Technical University of Eindhoven, the Netherlands. He likes to play games. INTERNATIONAL MATHEMATICS RESEARCH PAPERS www.imrp.oxfordjournals.org INTERNATIONAL MATHEMATICS RESEARCH SURVEYS www.imrs.oxfordjournals.org Portugaliae Mathematica A journal of the Portuguese Mathematical Society ISSN 0032-5155. 2008. Vol 65, 4 issues Approx. 500 pages. 17.0 x 24.0 cm Price of subscription, including electronic edition: 196 Euro plus shipping (add 20 Euro for normal delivery).Other subscriptions on request. Editor-in-Chief: Rui Loja Fernandes (Portugal) Executive Editorial Board: José Ferreira Alves, Miguel Ramos, Luis Nunes Vicente (all Portugal) Aims and Scope: Since its foundation in 1937, Portugaliae Mathematica has aimed at publishing high-level research articles in all branches of mathematics. With great efforts by its founders, the journal was able to publish articles by some of the best mathematicians of the time. In 2001 a New Series of Portugaliae Mathematica was started, reaffirming the purpose of maintaining a high-level research journal in mathematics with a wide range scope. Associate Editors: Jorge Almeida (Portugal), Michèle Audin (France), Hugo Beirão da Veiga (Italy), Eric A. Carlen (USA), João Paulo Dias (Portugal), Irene Fonseca (USA), Arnaldo Garcia (Brasil), Bernt Øksendal (Norway), Peter Olver (USA), Rafael Ortega (Spain), Rahul Pandharipande (USA), Alfio Quarteroni (Switzerland), José Francisco Rodrigues (Portugal), Bernd Sturmfels (USA), Susanna Terracini (Italy), Philippe Toint (Belgium), Marcelo Viana (Brazil), Alan Weinstein (USA), Enrique Zuazua (Spain). APPLIED MATHEMATICS RESEARCH EXPRESS www.amrx.oxfordjournals.org giving you faster access to the most recent research in mathematics European Mathematical Society Publishing House Seminar for Applied Mathematics, ETH-Zentrum FLI C4, Fliederstrasse 23 CH-8092 Zürich, Switzerland subscriptions@ems-ph.org www.ems-ph.org EMS Newsletter December 2007 17
Page 1 and 2: NEWSLETTER OF THE EUROPEAN MATHEMAT
Page 3 and 4: Editorial Team Editor-in-Chief Mart
Page 5 and 6: 5ECM: 14-18 July 2008 in Amsterdam
Page 7: Editorial The three Science lecture
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