LUNCH with a colleague from work should be a time to unwind - the most taxing task being to decide what to eat, drink and choose for dessert. For Rick Mabry and Paul Deiermann it has never been that simple. They can't think about sharing a pizza, for example, without falling headlong into the mathematics of how to slice it up. "We went to lunch together at least once a week," says Mabry, recalling the early 1990s when they were both at Louisiana State University, Shreveport. "One of us would bring a notebook, and we'd draw pictures while our food was getting cold."

    The problem that bothered them was this. Suppose the harried waiter cuts the pizza off-centre, but with all the edge-to-edge cuts crossing at a single point, and with the same angle between adjacent cuts. The off-centre cuts mean the slices will not all be the same size, so if two people take turns to take neighbouring slices, will they get equal shares by the time they have gone right round the pizza - and if not, who will get more?

    Of course you could estimate the area of each slice, tot them all up and work out each person's total from that. But these guys are mathematicians, and so that wouldn't quite do. They wanted to be able to distil the problem down to a few general, provable rules that avoid exact calculations, and that work every time for any circular pizza.

    As with many mathematical conundrums, the answer has arrived in stages - each looking at different possible cases of the problem. The easiest example to consider is when at least one cut passes plumb through the centre of the pizza. A quick sketch shows that the pieces then pair up on either side of the cut through the centre, and so can be divided evenly between the two diners, no matter how many cuts there are.

    So far so good, but what if none of the cuts passes through the centre? For a pizza cut once, the answer is obvious by inspection: whoever eats the centre eats more. The case of a pizza cut twice, yielding four slices, shows the same result: the person who eats the slice that contains the centre gets the bigger portion. That turns out to be an anomaly to the three general rules that deal with greater numbers of cuts, which would emerge over subsequent years to form the complete pizza theorem.

    The first proposes that if you cut a pizza through the chosen point with an even number of cuts more than 2, the pizza will be divided evenly between two diners who each take alternate slices. This side of the problem was first explored in 1967 by one L. J. Upton in Mathematics Magazine （vol 40, p 163）. Upton didn't bother with two cuts: he asked readers to prove that in the case of four cuts （making eight slices） the diners can share the pizza equally. Next came the general solution for an even number of cuts greater than 4, which first turned up as an answer to Upton's challenge in 1968, with elementary algebraic calculations of the exact area of the different slices revealing that, again, the pizza is always divided equally between the two diners （Mathematics Magazine, vol 41, p 46）.

    With an odd number of cuts, things start to get more complicated. Here the pizza theorem says that if you cut the pizza with 3, 7, 11, 15… cuts, and no cut goes through the centre, then the person who gets the slice that includes the centre of the pizza eats more in total. If you use 5, 9, 13, 17… cuts, the person who gets the centre ends up with less （see diagram）.

    Rigorously proving this to be true, however, has been a tough nut to crack. So difficult, in fact, that Mabry and Deiermann have only just finalised a proof that covers all possible cases.

    Their quest started in 1994, when Deiermann showed Mabry a revised version of the pizza problem, again published in Mathematics Magazine （vol 67, p 304）. Readers were invited to prove two specific cases of the pizza theorem. First, that if a pizza is cut three times （into six slices）, the person who eats the slice containing the pizza's centre eats more. Second, that if the pizza is cut five times （making 10 slices）, the opposite is true and the person who eats the centre eats less.

    The first statement was posed as a teaser: it had already been proved by the authors. The second statement, however, was preceded by an asterisk - a tiny symbol which, in Mathematics Magazine, can mean big trouble. It indicates that the proposers haven't yet proved the proposition themselves. "Perhaps most mathematicians would have thought, 'If those guys can't solve it, I'm not going to look at it.'" Mabry says. "We were stupid enough to look at it."

    Most mathematicians would have thought, 'I'm not going to look at it.' We were stupid enough to try

    Deiermann quickly sketched a solution to the three-cut problem - "one of the most clever things I've ever seen," as Mabry recalls. The pair went on to prove the statement for five cuts - even though new tangles emerged in the process - and then proved that if you cut the pizza seven times, you get the same result as for three cuts: the person who eats the centre of the pizza ends up with more.

    Boosted by their success, they thought they might have stumbled across a technique that could prove the entire pizza theorem once and for all. For an odd number of cuts, opposing slices inevitably go to different diners, so an intuitive solution is to simply compare the sizes of opposing slices and figure out who gets more, and by how much, before moving on to the next pair. Working your way around the pizza pan, you tot up the differences and there's your answer.

    Simple enough in principle, but it turned out to be horribly difficult in practice to come up with a solution that covered all the possible numbers of odd cuts. Mabry and Deiermann hoped they might be able to deploy a deft geometrical trick to simplify the problem. The key was the area of the rectangular strips lying between each cut and a parallel line passing through the centre of the pizza （see diagram）. That's because the difference in area between two opposing slices can be easily expressed in terms of the areas of the rectangular strips defined by the cuts. "The formula for [the area of] strips is easier than for slices," Mabry says. "And the strips give some very nice visual proofs of certain aspects of the problem."

    Unfortunately, the solution still included a complicated set of sums of algebraic series involving tricky powers of trigonometric functions. The expression was ugly, and even though Mabry and Deiermann didn't have to calculate the total exactly, they still had to prove it was positive or negative to find out who gets the bigger portion. It turned out to be a massive hurdle. "It ultimately took 11 years to figure that out," says Mabry.

    Over the following years, the pair returned occasionally to the pizza problem, but with only limited success. The breakthrough came in 2006, when Mabry was on a vacation in Kempten im Allg?u in the far south of Germany. "I had a nice hotel room, a nice cool environment, and no computer," he says. "I started thinking about it again, and that's when it all started working." Mabry and Deiermann - who by now was at Southeast Missouri State University in Cape Girardeau - had been using computer programs to test their results, but it wasn't until Mabry put the technology aside that he saw the problem clearly. He managed to refashion the algebra into a manageable, more elegant form.

    Back home, he put computer technology to work again. He suspected that someone, somewhere must already have worked out the simple-looking sums at the heart of the new expression, so he trawled the online world for theorems in the vast field of combinatorics - an area of pure mathematics concerned with listing, counting and rearranging - that might provide the key result he was looking for.

    Eventually he found what he was after: a 1999 paper that referenced a mathematical statement from 1979. There, Mabry found the tools he and Deiermann needed to show whether the complex algebra of the rectangular strips came out positive or negative. The rest of the proof then fell into place （American Mathematical Monthly, vol 116, p 423）.

    So, with the pizza theorem proved, will all kinds of important practical problems now be easier to deal with? In fact there don't seem to be any such applications - not that Mabry is unduly upset. "It's a funny thing about some mathematicians," he says. "We often don't care if the results have applications because the results are themselves so pretty." Sometimes these solutions to abstract mathematical problems do show their face in unexpected places. For example, a 19th-century mathematical curiosity called the "space-filling curve" - a sort of early fractal curve - recently resurfaced as a model for the shape of the human genome.

    Mabry and Deiermann have gone on to examine a host of other pizza-related problems. Who gets more crust, for example, and who will eat the most cheese? And what happens if the pizza is square? Equally appetising to the mathematical mind is the question of what happens if you add extra dimensions to the pizza. A three-dimensional pizza, one might argue, is a calzone - a bread pocket filled with pizza toppings - suggesting a whole host of calzone conjectures, many of which Mabry and Deiermann have already proved. It's a passion that has become increasingly theoretical over the years. So if on your next trip to a pizza joint you see someone scribbling formulae on a napkin, it's probably not Mabry. "This may ruin any pizza endorsements I ever hoped to get," he says, "but I don't eat much American pizza these days."

    工作后与同事共进午餐应该是一个放松的时刻-最费神的是要决定吃什么、喝什么以及选择甜点。对Rick和Paul Deiermann来说，这从来不是那么简单的。例如，如果他们没有仓促地陷入怎样切一块披萨的数学问题，他们是不会考虑共享一块披萨的。Mabry回忆起他们都在路易斯安那州大学的时候，说："我们至少共进午餐一周一次，我们俩会有一个带着笔记本，我们在画各种图形，而食物已经变凉了。"

    使他们迷惑的问题是这样的。假设急匆匆的服务生是从偏离中心的位置切一块披萨的，但是所有边-到-边的切线（即切披萨的线）都相交于一点，且相邻切线间的角度是相等的。偏离中心的切法意味着披萨片的大小是不同的，因此如果两个人轮流按顺序依次拿披萨直到他们分完，那么他们会得到相同量的披萨吗？如果不会，谁拿到的更多？

    当然你可以估计每一块的面积，把面积加起来得到每个人拿到的总面积。但是这两个人是数学家，所以他们不会这样做。他们希望能够把这一问题的实质归纳成几条普遍的、可证明的定理，以避免精确的计算，并希望只要是圆形的披萨，这些定理都适用。

    和许多数学上的谜题一样，这一问题登上了舞台-每个人都在寻找不同的可能的情况。最简单的例子是考虑什么时候至少有一刀是恰好经过披萨中心的。一种快速粗略的想法是披萨片是沿着经过中心的那一刀成对分布的，因此无论切多少刀，两个人都能吃到同量的披萨。

    要是这样就好了。如果没有一刀是经过中心的呢？对于只切一刀的披萨，问题很明显：谁吃到了中心，谁就吃得多。切两刀分成四块的情况表明同样的结果：吃到含有中心那块披萨的人得到更多。但当切更多的刀时，这证明是违反了三个定理，这一问题出现在以后的很多年里，形成了完整的披萨定理。

    第一个人提出，如果通过你选择的一点切一块披萨，刀数是大于2的偶数，那么披萨会在两个用餐者之间平均分配，如果两个人是轮流吃的话。1967年，一个叫L.J.Upton的人在《数学》杂志上首次探讨了这一方面，他没有为刀数为2时的情况费心：他要求读者去证明切四刀时（八块披萨）,两个人仍能平均分享披萨。接下来对于大于四刀的偶数，出现了问题的通解。1968年，Upton的问题首次得到解答，答案使用基本的代数计算算出了不同披萨片的精确面积，它表明，披萨总是能够在两个人中间平均分配。

    如果刀数为奇数，问题变得更加复杂。披萨定理认为如果你分别用3、7、11、15刀来切，且没有一刀是经过中心的，那么吃到有中心披萨片的人吃得多。如果你用5、9、13、17刀来切，吃到有中心披萨片的人吃得少。

    然而要严格证明这个理论却非易事。事实上，它是如此困难以至于Marby和Deiermann只能用一种包含所有可能情况的证明来定稿。

    Marby和Deierman对这一问题的探求始于1994年，当时Deiermann给Mabry看了披萨问题的修订版，并再一次刊登在《数学杂志》上。读者们被邀请来证明披萨定理的两种特例。首先，如果披萨被切了三次（六块）,吃到有中心披萨片的人吃得多。其次，如果披萨被切了五次（十块）,吃到有中心披萨片的人吃得少。

    第一种观点是用来抛砖引玉的：它早已被作者证明过。而第二种观点前面加了一个星号-在《数学杂志》上，这一小符号代表了一个大问题。它表明，提出者本人还没有办法证明他们提出的观点。"也许大多数数学家已经想过，如果他们都不能解决，那我将放弃研究它，"Marby说。"去解决这个问题，我们已经够蠢了。"

    Dieermann对三刀问题的答案快速列了草图，Marby回忆说"是我见过的最聪明的事情之一。"他们继续证明了五刀切的理论-尽管在过程中又出现了新的难题-然后证明了七刀切的理论，如果你对一块披萨切七次，你将得到与切三次相同的结果，即吃到含有中心的披萨片的人吃得更多。

    受到成功的鼓舞，他们认为也许他们偶然发现了一种技术，这种技术能一劳永逸地证明整个披萨定理。对于刀数为奇数的切法，相对的披萨片不可避免地被不同的人所食用，因此一种直观的解决方法是简单地比较相对两块披萨片的大小，然后计算出谁吃得多，然后比较下一对披萨片的大小。当披萨的一整圈都轮完了，你就可以把结果加起来，得到结果了。

    理论上很简单，但要提出一种方法来概括刀数为偶数时所有可能的情况，实际上困难得多。Mabry和Deiermann希望他们可以用一种简洁的几何方法把问题简化。问题的关键是在每一刀之间的长方形以及与穿过中心线平行的线。那是因为相对的两块披萨面积的大小可以用长方形的面积来表示。"长方形的面积公式比披萨的简单得多。"Marby说："并且长方形给出了这一问题有关方面的直观证据。"

    不幸的是，这一方法仍然包含了一系列复杂的代数计算，还涉及了复杂得三角函数。这个表达式令人头痛，尽管如此，他们还是不得不计算出精确结果，他们仍要证明谁得到更多披萨的观点是正确的还是错误的。结果证明这是一个巨大的障碍。"最终耗费了11年才弄清楚",Marby说。

    在接下来的几年里，两个人偶尔会讨论一下披萨问题，但是只有有限的进展。2006年，问题终于有了突破，此时Mabry正在法国极靠南的Kempten渡假。"我住在一个很好的旅馆房间里，舒服凉爽的环境，没有电脑，"他说"我再一次开始想这个问题，就是那时所有一切都想通了。"此前，Mabry和Deiermann在东南部的密苏里州大学，一直用计算机程序检验他们的结果。但是，直到Mabry放下了计算机技术，问题才迎刃而解。他成功地把代数公式改进成了更易处理、更简洁美观的形式。

    回家后，他又用计算机开始了工作。他怀疑有人在其他地方已经就计算出了这种结果看起来很简单的形式，可能存在于一些新表达式中，因此他去网上搜索，大范围中组合起来的各种关键词-一种只有在数学中才用到的方法，涉及列表、计算和重排-这可能能使他找到一直在寻找的结果。

    最终他找到了他想要的：一篇1999年的论文，引用了一个1979得数学观点。在那里，他找到了他们需要的工具，用这个工具可以说明长方形面积的复杂代数公式是正确的还是错误的。剩下的证据一一得到了证明。

    因此，随着披萨定理被证明了，那么一些重要的实际问题就能更容易地解决了吗？事实上，人们还看不到披萨定理会有什么应用-并不是Mabry过分悲观了。他说；"对数学家来说，这是一个有趣的问题，我们通常不关心结果是否能有应用因为结果本事就很完美。"有时，抽象数学问题的解答确实会在意想不到的领域中得到应用。例如，19世纪一个数学家的好奇心-叫做"空间-充满曲线"-一种早期的分形曲线-最近重新浮出水面，作为模拟人类基因组形状的模型。

    Mabry和Deiermann继续检验了一系列其他的关于披萨的问题。例如，谁会吃到更多的披萨皮？谁会吃到更多的奶酪？如果披萨是方形的，情况又该如何呢？如果增加了维数情况又会怎样，这同样引起数学家的兴趣。一个三维的披萨，是一个半圆形的烤馅饼，一个充满各种披萨配料的面包袋，它又会引出一系列关于半圆形的猜想，其中的一些已经被Mabry和Deiermann证明了。它是一种热情，多年里渐渐变成了一种理论。如果下次你去吃披萨，看到某个人在纸巾上涂写公式，那一定不是Mabry."虽然会破坏我曾经希望得到的披萨定理，但我这些日子确实不再吃很多美国披萨了。"