星报通讯社电/“数独”游戏(Sudoku)有不少的痴迷者，但也有许多局外人常常笑他们自讨苦吃，在其难无比的数字游戏上浪费时间。可是，最近皇后大学(Queen’s University)的2名教授发表论文指出，玩“数独”游戏不仅有助提高数学技能，还对纯数学研究提出了新的挑战。
安省皇后大学教授何姿珀(Agnes Herzberg)和墨提(Ram Murty)都是“数独”迷，在发现彼此都醉心与这个风靡的游戏时，决定运用2人的学术才智，联手揭开“数独”的谜底。
图着色问题
身为数字理论专家的墨提表示，玩“数独”，可能没有料到他们所从事的其实是数学家所称的图着色问题。人们玩“数独”游戏所需要培养的逻辑技能，对数学中的清晰思维至关重要。
墨提和统计学专家何姿珀运用纯数学的一个深奥分支图着色理论，首次定立了“数独”游戏只得出一个正确解法的最低前提。他们表示，两人都发现过报纸和杂志上的“数独”游戏存在至少2种正确的解法。
何姿珀表示，该游戏的本意是只有一种解法，但事实并非都如此。他们的研究成果今天由美国数学协会(American Mathematical Society)发表，其论文显示，该游戏必须有至少17个起头条目，带有至少8个不同数字，才可能做到只有一个答案，而拥有多达29个起头条目的游戏，仍然可能出现超过一个的答案。他们还发现，总共存在5,472,730,538个不同但有效的“数独”游戏。
为了揭开“数独”的谜底，他们把81个方格当作一个图谱上的点，每个都可能是9种不同颜色之一。但共边的两个点不能是相同的颜色，正如相同的数字不能出现在两行、列或九乘九格“数独”游戏的内方框上。
纯数学已研究出了进行图着色的先进方法，两名教授将其成功地运用到了“数独”上。
但是，他们对“数独”的一些特点仍然一筹莫展，例如能得出唯一答案的起头数字的绝对最小值。墨提表示，它可能是16或更小，但电脑也无法对此作出必要的运算。
还有，两人都称对“数独”创造者如何为他们的游戏定出难度大惑不解，因为有时“魔鬼难度”的游戏反而比“容易”的更好解。大多数的“数独”游戏皆由电脑程序生成，但一些仍由手工编制，像填词游戏一样。
可能用于敏感资料
墨提表示，“数独”研究可能实际运用于敏感资料–例如金融资料的转移 –的加密上。如果网格上17个数字形成一个独一无二的答案，那么由所有81个数字所代表的讯息只要发送出这17个数字，即可全部通过，这就防止了可能的中途拦截。
数字理论是纯数学的一个分支，统而言之，研究的是数字的特性，重点研究整数(1，2，3…)。数字理论的基础是质数(只能被1和本身整除的整数)的分布和特性。
Sudoku helps hone logic
Professors say Sudoku helps players hone logic skills – `You could call it math by stealth’
Jun 07, 2007 04:30 AM
Peter Calamai
Science writer
Sudoku addicts can now scoff at sourpusses who complain they’re wasting time on those often fiendish numbers puzzles.
Two Queen’s University professors say instead that tackling Sudoku is fostering better math skills among Canadians and providing new challenges for academics who specialize in pure mathematics.
Sudoku enthusiasts Agnes Herzberg and Ram Murty pooled their respective academic talents to tackle the mysteries of Sudoku after discovering both were fascinated by the phenomenal popularity of the puzzles.
“If you told Sudoku aficionados that they’re actually doing something mathematicians call a graph theory colouring problem, they’d be bowled over,” says Murty, a specialist in number theory. Herzberg is an expert in statistics.
Yet the logical skills people must hone to solve Sudoku puzzles are crucial to thinking clearly about mathematics, he said.
“You could call it math by stealth,” he said.
Applying the arcane branch of pure mathematics known as graph colouring allowed the Queen’s professors to lay down for the first time ever the minimal conditions for a Sudoku puzzle to have just one correct solution.
Both Murty and Herzberg say they’ve come across Sudokus in newspapers and magazines that had at least two correct solutions. They use one from Air Canada’s enRoute magazine in their research paper.
“The implication is that you do the puzzle and there is only one answer. But that’s not always the case,” said Herzberg.
Their research, published today by the American Mathematical Society, shows that a puzzle must start with at least 17 entries with a minimum of eight different numbers to offer the prospect of just one answer and puzzles with as many as 29 starting entries can still have more than one correct solution.
And there are more than enough squares to feed the obsession: the researchers say there are 5,472,730,538 different and valid Sudoku games.
They tackled the mysteries of Sudoku by treating the 81 squares as points on a graph each of which could be one of nine different colours. But pairs of these points which shared an edge couldn’t be the identical colour, just as the same number can’t be in two rows, columns or inner squares of the nine-by-nine Sudoku puzzle.
Pure mathematics has developed sophisticated approaches to graph colouring which the Queen’s professors successfully applied to Sudokus.
But the professors are still stumped by some Sudoku characteristics, like the absolute minimum number of initial numbers that will produce a puzzle with only one solution. It could be 16 or even smaller, says Murty, and computers simply aren’t up to the job of running the necessary calculation.
As well, both say they’re puzzled by how Sudoku creators come up with the difficulty ratings for their puzzles, with “fiendish” sometimes proving less trouble than “easy.” Most Sudokus are generated by computer programs, although some are still handcrafted like crossword puzzles.
Murty says the Sudoku research could have practical applications in the encryption of sensitive information, such as financial transfers. If 17 numbers in a grid produce a unique answer, then a message represented by all 81 numbers could be passed along by transmitting only those 17, foiling would-be interception.
