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."

    工作后與同事共進午餐應該是一個放松的時刻-最費神的是要決定吃什么、喝什么以及選擇甜點。對Rick和Paul Deiermann來說，這從來不是那么簡單的。例如，如果他們沒有倉促地陷入怎樣切一塊披薩的數學問題，他們是不會考慮共享一塊披薩的。Mabry回憶起他們都在路易斯安那州大學的時候，說："我們至少共進午餐一周一次，我們倆會有一個帶著筆記本，我們在畫各種圖形，而食物已經變涼了。"

    使他們迷惑的問題是這樣的。假設急匆匆的服務生是從偏離中心的位置切一塊披薩的，但是所有邊-到-邊的切線（即切披薩的線）都相交于一點，且相鄰切線間的角度是相等的。偏離中心的切法意味著披薩片的大小是不同的，因此如果兩個人輪流按順序依次拿披薩直到他們分完，那么他們會得到相同量的披薩嗎？如果不會，誰拿到的更多？

    當然你可以估計每一塊的面積，把面積加起來得到每個人拿到的總面積。但是這兩個人是數學家，所以他們不會這樣做。他們希望能夠把這一問題的實質歸納成幾條普遍的、可證明的定理，以避免精確的計算，并希望只要是圓形的披薩，這些定理都適用。

    和許多數學上的謎題一樣，這一問題登上了舞臺-每個人都在尋找不同的可能的情況。最簡單的例子是考慮什么時候至少有一刀是恰好經過披薩中心的。一種快速粗略的想法是披薩片是沿著經過中心的那一刀成對分布的，因此無論切多少刀，兩個人都能吃到同量的披薩。

    要是這樣就好了。如果沒有一刀是經過中心的呢？對于只切一刀的披薩，問題很明顯：誰吃到了中心，誰就吃得多。切兩刀分成四塊的情況表明同樣的結果：吃到含有中心那塊披薩的人得到更多。但當切更多的刀時，這證明是違反了三個定理，這一問題出現在以后的很多年里，形成了完整的披薩定理。

    第一個人提出，如果通過你選擇的一點切一塊披薩，刀數是大于2的偶數，那么披薩會在兩個用餐者之間平均分配，如果兩個人是輪流吃的話。1967年，一個叫L.J.Upton的人在《數學》雜志上首次探討了這一方面，他沒有為刀數為2時的情況費心：他要求讀者去證明切四刀時（八塊披薩）,兩個人仍能平均分享披薩。接下來對于大于四刀的偶數，出現了問題的通解。1968年，Upton的問題首次得到解答，答案使用基本的代數計算算出了不同披薩片的精確面積，它表明，披薩總是能夠在兩個人中間平均分配。

    如果刀數為奇數，問題變得更加復雜。披薩定理認為如果你分別用3、7、11、15刀來切，且沒有一刀是經過中心的，那么吃到有中心披薩片的人吃得多。如果你用5、9、13、17刀來切，吃到有中心披薩片的人吃得少。

    然而要嚴格證明這個理論卻非易事。事實上，它是如此困難以至于Marby和Deiermann只能用一種包含所有可能情況的證明來定稿。

    Marby和Deierman對這一問題的探求始于1994年，當時Deiermann給Mabry看了披薩問題的修訂版，并再一次刊登在《數學雜志》上。讀者們被邀請來證明披薩定理的兩種特例。首先，如果披薩被切了三次（六塊）,吃到有中心披薩片的人吃得多。其次，如果披薩被切了五次（十塊）,吃到有中心披薩片的人吃得少。

    第一種觀點是用來拋磚引玉的：它早已被作者證明過。而第二種觀點前面加了一個星號-在《數學雜志》上，這一小符號代表了一個大問題。它表明，提出者本人還沒有辦法證明他們提出的觀點。"也許大多數數學家已經想過，如果他們都不能解決，那我將放棄研究它，"Marby說。"去解決這個問題，我們已經夠蠢了。"

    Dieermann對三刀問題的答案快速列了草圖，Marby回憶說"是我見過的最聰明的事情之一。"他們繼續證明了五刀切的理論-盡管在過程中又出現了新的難題-然后證明了七刀切的理論，如果你對一塊披薩切七次，你將得到與切三次相同的結果，即吃到含有中心的披薩片的人吃得更多。

    受到成功的鼓舞，他們認為也許他們偶然發現了一種技術，這種技術能一勞永逸地證明整個披薩定理。對于刀數為奇數的切法，相對的披薩片不可避免地被不同的人所食用，因此一種直觀的解決方法是簡單地比較相對兩塊披薩片的大小，然后計算出誰吃得多，然后比較下一對披薩片的大小。當披薩的一整圈都輪完了，你就可以把結果加起來，得到結果了。

    理論上很簡單，但要提出一種方法來概括刀數為偶數時所有可能的情況，實際上困難得多。Mabry和Deiermann希望他們可以用一種簡潔的幾何方法把問題簡化。問題的關鍵是在每一刀之間的長方形以及與穿過中心線平行的線。那是因為相對的兩塊披薩面積的大小可以用長方形的面積來表示。"長方形的面積公式比披薩的簡單得多。"Marby說："并且長方形給出了這一問題有關方面的直觀證據。"

    不幸的是，這一方法仍然包含了一系列復雜的代數計算，還涉及了復雜得三角函數。這個表達式令人頭痛，盡管如此，他們還是不得不計算出精確結果，他們仍要證明誰得到更多披薩的觀點是正確的還是錯誤的。結果證明這是一個巨大的障礙。"最終耗費了11年才弄清楚",Marby說。

    在接下來的幾年里，兩個人偶爾會討論一下披薩問題，但是只有有限的進展。2006年，問題終于有了突破，此時Mabry正在法國極靠南的Kempten渡假。"我住在一個很好的旅館房間里，舒服涼爽的環境，沒有電腦，"他說"我再一次開始想這個問題，就是那時所有一切都想通了。"此前，Mabry和Deiermann在東南部的密蘇里州大學，一直用計算機程序檢驗他們的結果。但是，直到Mabry放下了計算機技術，問題才迎刃而解。他成功地把代數公式改進成了更易處理、更簡潔美觀的形式。

    回家后，他又用計算機開始了工作。他懷疑有人在其他地方已經就計算出了這種結果看起來很簡單的形式，可能存在于一些新表達式中，因此他去網上搜索，大范圍中組合起來的各種關鍵詞-一種只有在數學中才用到的方法，涉及列表、計算和重排-這可能能使他找到一直在尋找的結果。

    最終他找到了他想要的：一篇1999年的論文，引用了一個1979得數學觀點。在那里，他找到了他們需要的工具，用這個工具可以說明長方形面積的復雜代數公式是正確的還是錯誤的。剩下的證據一一得到了證明。

    因此，隨著披薩定理被證明了，那么一些重要的實際問題就能更容易地解決了嗎？事實上，人們還看不到披薩定理會有什么應用-并不是Mabry過分悲觀了。他說；"對數學家來說，這是一個有趣的問題，我們通常不關心結果是否能有應用因為結果本事就很完美。"有時，抽象數學問題的解答確實會在意想不到的領域中得到應用。例如，19世紀一個數學家的好奇心-叫做"空間-充滿曲線"-一種早期的分形曲線-最近重新浮出水面，作為模擬人類基因組形狀的模型。

    Mabry和Deiermann繼續檢驗了一系列其他的關于披薩的問題。例如，誰會吃到更多的披薩皮？誰會吃到更多的奶酪？如果披薩是方形的，情況又該如何呢？如果增加了維數情況又會怎樣，這同樣引起數學家的興趣。一個三維的披薩，是一個半圓形的烤餡餅，一個充滿各種披薩配料的面包袋，它又會引出一系列關于半圓形的猜想，其中的一些已經被Mabry和Deiermann證明了。它是一種熱情，多年里漸漸變成了一種理論。如果下次你去吃披薩，看到某個人在紙巾上涂寫公式，那一定不是Mabry."雖然會破壞我曾經希望得到的披薩定理，但我這些日子確實不再吃很多美國披薩了。"