Issue 023, 2022 SCIENCE FOCUS Peto’s Paradox: Is Body Size the Key to Fighting Cancer? 佩托悖論：體型大小是對抗癌症的關鍵嗎？ Not One Less: Fencepost Errors 一個都不能遺漏：柵欄錯誤 The Number Sequence That Needs to Be Said 唸出數列之必要：聽射性數列 Haber Process: More Than Just Nitrogen and Ammonia 哈柏法：改變世界的化學反應 Will Mathematicians Be Replaced? – Computers in Mathematics 數學家會被取代嗎？—電腦在數學中的角色
Dear Readers, Welcome to the new issue of Science Focus. In addition to being a source of your “book reports”, I hope this magazine can serve as a starting point for your further exploration of specific topics. When our writers prepare each article, they normally consult multiple references to ensure the accuracy of the content. I hope you will develop the same habit during your personal journey of discovery. Did you know that elephants are much less likely to suffer from cancer than us? Find out why from our cover story on Peto’s paradox when we discussed the surprising lack of correlation between body size and cancer. One way to treat cancer is to undergo surgery. It turns out we have Joseph Lister to thank for popularizing antiseptic surgeries, which greatly increase the survival rate of patients. Turning to mathematics, we consider a seemingly simple quest of counting objects or birthdays and how it can go wrong. We will also explore how the concept of radioactive decay can be applied to a mathematical problem. For those of you who enjoy horticulture, we revisit how fertilizers are made using the Haber process. Is there more to learn beyond the chemistry textbook? Finally, don’t miss our articles on the forefront of scientific breakthroughs, from using light to control brain activities to using artificial intelligence for mathematical proofs. As always, please follow our social media channels for additional bite-size scientific stories. Drop us a line in the comments section. We would love to hear from you! Yours faithfully, Prof. Ho Yi Mak Editor-in-Chief 親愛的讀者： 歡迎閱讀最新一期《科言》！除了可能作為閱讀報告的「指定書目」 外，我希望這本雜誌能為你提供探索科學題材的切入點。當我們的學生 編輯預備每篇文章時，他們通常都會參考多篇文獻以確保內容的準確 性，希望大家踏上科學旅程時也能養成同樣習慣。 你知道大象患癌的機率遠比我們小嗎？箇中原因可以在關於佩托 悖論的封面故事中找到，我們會討論體型和癌症背後出乎意料地沒有 關聯的原因。進行手術是治療癌症的其中一個方法，說到手術，我們要 感謝 Joseph Lister 使抗菌程序普及化，這大大增加了病人手術後的存 活率。也讓我們來談談數學，看看我們如何在的數算物件或生日的過程 中出亂子，亦窺探一下數學家如何把放射性衰變的概念應用在數學問題 上。此外，我們亦為喜歡園藝的你重溫一下透過哈柏法製造肥料的過程， 教科書以外還有沒有甚麼是值得學習的呢？最後，不要錯過介紹最新科 學突破的文章，當中包括運用光來控制腦部活動和利用人工智能來寫數 學證明的技術。 一如以往，我們歡迎大家追蹤我們的社交專頁，那裡會連載一些雜 誌以外的科學小故事。我們亦期待在網上與大家互動，歡迎隨時留言給 我們！ 主編 麥晧怡教授 敬上 Message from the Editor-in-Chief 主編的話 Copyright © 2022 HKUST E-mail: sciencefocus@ust.hk Homepage: http://sciencefocus.ust.hk Scientific Advisors 科學顧問 Prof. Jason Chan陳鈞傑教授 Prof. Ivan Ip 葉智皓教授 Prof. Stanley Lau 劉振鈞教授 Prof. Tim Leung 梁承裕教授 Editor-in-Chief 主編輯 Prof. Ho Yi Mak麥晧怡教授 Managing Editor 總編輯 Daniel Lau 劉劭行 Student Editorial Board學生編委 Editors 編輯 Sonia Choy 蔡蒨珩 Peace Foo 胡適之 Dana Kim 金娥凜 Sirius Lee 李揚 Lambert Leung 梁卓霖 Graphic Designers 設計師 Tiffany Kwok 郭喬 Charley Lam 林曉薏 Samantha Kaitlyn Ng Cheuk Hei Tsang 曾卓希 Contents Science Focus Issue 023, 2022 What’s Happening in Hong Kong? 香港科技活動 Dinosaurs of Antarctica 3D 1 極地尋龍 3D The Shaw Prize 2022 Exhibition 2022 邵逸夫獎展覽 Amusing World of Science 趣味科學 Peto’s Paradox: Is Body Size the Key to Fighting Cancer? 2 佩托悖論：體型大小是對抗癌症的關鍵嗎？ Not One Less: Fencepost Errors 6 一個都不能遺漏：柵欄錯誤 The Number Sequence That Needs to Be Said 10 唸出數列之必要：聽射性數列 Science in History 昔日科學 The Father of Antiseptic Surgery: Joseph Lister 14 抗菌手術之父：Joseph Lister Science Today 今日科學 Haber Process: More Than Just Nitrogen and Ammonia 16 哈柏法：改變世界的化學反應 Optogenetics: Controlling Brain Activities with Light 20 光遺傳學：運用光控制腦部活動 Will Mathematicians Be Replaced? – 22 Computers in Mathematics 數學家會被取代嗎？—電腦在數學中的角色
What’s Happening in Hong Kong? 香港科技活動 Fun in Fall Science Activities 秋日科學好節目 Any plans for this fall? Check out these activities! 計劃好這個秋天的好去處了嗎？不妨考慮以下活動！ Dinosaurs of Antarctica 3D 極地尋龍3D The Shaw Prize 2022 Exhibition 2022邵逸夫獎展覽 The now frozen Antarctica was once warm and forested hundreds of millions of years ago with dinosaurs roaming freely. There was fierce competition for survival on the continent. Let’s follow a team of paleoecologists to examine the fossi ls of those prehistoric dinosaurs and witness the dramatic transformation of the southern continent through this 3D dome show which utilized next-level computer graphics to reconstruct the continent’s hidden-greener-past. Established in 2002, the Shaw Prize recognizes currently active scientists with recent significant breakthroughs in scientific research. Three prizes are awarded annually in Mathematical Sciences, Life Science and Medicine, and Astronomy. In the exhibition, you can learn more about the Shaw Laureates of this year and their scientific achievements. Show period: July 1, 2022 – March 31, 2023 Time: 2:00 PMand 6:30 PMonMonday, Wednesday, Thursday and Friday (except public holiday) 12:30 PM and 5:00 PM on Saturday, Sunday and public holiday Venue: Space Theatre, Hong Kong SpaceMuseum Admission fee: Standard admission: $32 (stalls), $24 (front stalls) Concession admission: $16 (stalls), $12 (front stalls) Remark: Please refer to the museum’s website for more details. Date: September 30, 2022 – November 30, 2022 Venue: G/F Exhibition Hall, Hong Kong Science Museum 展期：2022年７月１日至 2023 年３月３1日 時間：星期一、三、四及五（公眾假期除外） 下午二時正及六時三十分 星期六、日及公眾假期下午十二時三十分 及五時正 地點：香港太空館天象廳 入場費：標準票：32元（後座）；24 元（前座） 優惠票：16 元（後座）；12 元（前座） 備註：更多詳情請參閱太空館網頁。 展期：2022年９月３０日至 2022 年１１月３０日 地點：香港科學館地下展覽廳 現時冰封的南極洲在數億年前曾是氣候溫 和的森林，有著無數恐龍在叢林裡自由遊走， 然而要在猛獸橫行的南極洲生存是非常艱辛的 任務。就讓我們藉著這套利用了頂尖電腦特效 去重塑南極洲過往翠綠面貌的 3D 球幕電影， 隨古生態學家去檢視那些史前恐龍的化石和見 證南極洲的巨大變化吧！ 邵逸夫獎於 2002 年設立，設有三個獎項： 數學科學獎、生命科學與醫學獎和天文學獎， 每年頒發予現時活躍於研究工作並在近期取得 突破性成果的科學家。透過今次展覽，你可以更 深入認識今年的得獎者以及他們的科研成就。 1
Cancer – a disease notorious for its deadliness. Till now, be it chemotherapy or immunotherapy, it still has no perfect cures. A sad truth indeed, but have you ever thought about cancers in other animals, like mice and elephants? Make a guess on the likelihood of these animals getting cancer – the answer may come as a surprise. By Lambert Leung 梁卓霖 F i r s t , how exact l y d o e s c a n c e r develop? In general, it can be caused by mutations i n p roto - oncogenes o r t umo r suppressor genes, both of which function to regulate normal cell growth and division. There are multiple cell cycle checkpoints to ensure that the genome i s proper l y repl icated. Tumor suppressor genes may kick in to repair damaged DNA, arrest cell cycle or induce apoptosis (cell suicide) when DNA replication goes wrong. However, when these genes are mutated, the cell may gain the ability to escape from the protective mechanisms and divide in an uncontrolled manner, forming a tumor. J udg i ng f r om t h i s mechan i sm, i t i s reasonable to deduce that larger organisms, which clear ly have more cel l s, are more prone to developing cancer because cell division has obviously occurred many more times in those creatures. Random mutations (mistakes in replication) may take place in every round of cel l division, so the chance of larger and older individuals having cel l s accumulat ing enough of those deadly mutations should be higher. However, just as organisms wi th 1,000 t imes more cel l s than humans don’t have an increased risk of developing cancer, we are not more cancer-prone than mice [1]. Such a lack of cor relation between body size and risk of
3 cancer is called the Peto’s paradox (footnote 1), named af ter the Engl i sh statistician and epidemiologist Richard Peto [2]. Simply terming the phenomenon as a paradox is not enough for scientists – currently, there are two general explanations for the anomaly, with the first one being genetics. The TP53 gene, which encodes p53 proteins, is one of the tumor suppressor genes. These proteins are located at the nuclei of cells, monitoring the DNA. When the DNA is damaged, p53 proteins will pause the cell cycle and activate other DNA repair genes if the damage is repai rable, otherwise cel l death will be induced to prevent further replication and perpetuation of the potentially harmful mutated DNA [3, 4]. Previous studies have demonst rated that loss-of-function mutations in TP53 (which in turn produces p53 proteins that are not fully functional) was found in over 50% of human cancers, suggesting the impo r tance of th i s mechan i sm i n cance r suppression [5]. Subjected to higher mutation r isks, elephants have evolved to contain 20 copies of the TP53 gene, whereas humans only have one copy [6]. This means the extra copies in elephants can compensate for mutated ones, retaining the ability to kill cancerous cells in the case of mutations. In contrast, if the only TP53 gene in humans is mutated, it can lead to an inheritable genetic condition called the Li-Fraumeni Syndrome, in which the individual is susceptible to a wide range of cancers at a young age [7]. In addition to the number of TP53 copies, a recent study also revealed that elephants had historically restored the function of an ancient, non-functioning gene remnant called LIF6. In response to DNA damage, LIF6 proteins can be activated by p53 proteins to effectively induce apoptosis and kill abnormal cells before they become cancerous [8]. This could be the reason why elephant can evol ve i n t o t h e o n l y paenungulate (footnote 2) with an exceptionally large body si ze whi le being resi l ient to cancer. Therefore, larger organisms could have unique tumor suppression mechanisms to genetically fight against cancer. Hypertumors, the tumors of tumors, are the second explanation of Peto’s paradox. Unl ike normal cel ls, cancerous ones are competitive and not cooperative in nature. This is characterized by tumor angiogenesis, the formation of blood vessels to provide ex t ra blood suppl y (wi th ox ygen and nutr ients) for prol iferation [9]. Based on this fact, it is not hard to imagine that any tumor would try to capture resources by any means. Research had predicted that the competing nature of cancer cells favor the formation of parasitic hypertumors, which feed on the parent tumor’s blood vessels [10]. Before a tumor can grow to a lethal size in large organisms, hypertumors would have emerged and stop the evil plan of the parent tumor by depriving their resources and keeping them at a sublethal size [10]. Nevertheless, this cannot be achieved in smaller organisms because a small tumor, which takes much less time for a single cancer cel l to develop to, is already l ifethreatening to the host. Hypertumors simply
癌症一直以其致命性臭名昭著，即使是化 療或免疫療法，今天依然沒有根治癌症的完美 方法。那固然是個沉重的事實，但你又可曾想 過癌症在老鼠和大象等動物身上又會是 怎樣的嗎？試猜猜牠們生癌的機 會，答案也許會出乎意料！ 回到最根本的問 題：癌症的成因 是甚麼？ 它 do not have enough time to develop. In a nutshell, there could be literally be cancers killing cancers in large organisms. With the cruel words of “there i s no t reatment”, the ear l iest descr ipt ion of cancer cal led the Edwin Smith Papyrus was found in Egypt in about 3000 B.C. [11]. Still being incurable at present, it is not an overstatement to say that cancer is the “king of disease”. The good of studying Peto’s paradox is that scientists may gain insights into how organisms cope with cancer and develop new therapeutic strategies. 1 Paradox: A contradiction that often goes against common sense. 2 Paenungulate: Members in the clade Paenungulata, which includes smaller creatures like hyrax and manatee (whose body size is still significantly smaller than that of an elephant) [8]. 普遍是由原癌基因或 腫瘤抑制基因突變所致，兩 者都有調節細胞正常生長和分裂 的功能。細胞週期裡有幾個檢查點確 保整個基因組被正確地複製，當 DNA 複製 過程出錯時，腫瘤抑制基因便會嘗試修復 DNA、 叫停細胞週期，甚至觸發細胞凋亡（即是促使細胞自 殺）。然而，如果上述兩種基因發生突變會令細胞喪失這種 保護機制，使它能避過監控不受控制地分裂，形成腫瘤。 從以上機制來看，我們可以推論出體型較大的生物生 癌的機會應該會較高，因為牠們體內有較多細胞，細胞分 裂的次數也明顯會遠比體型較小的生物多。由於每次細胞 分裂都有可能出現隨機的基因突變（即是複製上的錯誤）， 因此體型較大和較年老的個體擁有這些累積了足夠有害突 變的細胞的機會也應該會更高。但事實上，細胞數量比人 類多1000 倍的生物並會不因此較易生癌，而我們生癌的 機會也不比小鼠高 [1]。體型和生癌機會沒有關係這個現象 被稱為「佩托悖論」（Peto’s paradox；註一），名稱以英 國統計學和流行病學家 Richard Peto 命名 [2]。 科學家當然不會止於把現象歸類為悖論後就罷休： 到目前為止，佩托悖論有兩種主流解釋，第一個與遺傳學 有關。TP53基因是其中一種腫瘤抑制基因，編碼著位於細 胞核負責監察 DNA 的 p53 蛋白。當 DNA 受損時，如果 是可以修復的損傷，p53 會煞停細胞週期並激活其他 DNA 修復基因；否則，它會觸發細胞凋亡以防細胞繼續複製已 突變的 DNA，避免可能有害的 DNA 繼續存在 [3, 4]。過 往研究在超過 50% 的癌症患者身上也發現TP53功能喪 失型突變（loss-of-function mutations；意指會使製造 出來的 p53 蛋白喪失完整功能的突變），暗示了這個保護 機制對抑制癌症的重要性 [5]。 雖然體積龐大，但大象不容易有癌症的其中一個原因 是因為它擁有多達 20 組TP53基因，而人類則只有一組 [6]，意味著即使大象體內的一些TP53基因發生突變，其 他正常的複本也能繼續對抗癌細胞；相比之下，人類唯一 的 TP53 基因發生突變會導致一種名為「李法美尼症候群」 （Li-Fraumeni Syndrome）的遺傳病，令患者容易在年輕 時就罹患多種癌症 [7]。除了TP53基因複本的數量外，近 年研究亦發現大象曾經在演化過程中使已失去功能的遠古
5 References 參考資料： [1] Caulin, A. F., & Maley, C. C. (2011). Peto’s Paradox: Evolution’s Prescription for Cancer Prevention. Trends in Ecology & Evolution, 26(4), 175–182. doi:10.1016/j.tree.2011.01.002 [2] Peto, R., Roe, F. J., Lee, P. N., Levy, L., & Clack, J. (1975). Cancer and ageing in mice and men. British Journal of Cancer, 32(4), 411–426. doi:10.1038/bjc.1975.242 [3] National Institute of Health. (2020). TP53 gene. Retrieved from https://medlineplus.gov/genetics/gene/tp53/ [4] Mathews, C., van Holde, K., Appling, D., & Anthony-Cahill, A. (2013). Biochemistry (4th ed.). Toronto: Pearson. [5] Ozaki, T., & Nakagawara, A. (2011). Role of p53 in Cell Death and Human Cancers. Cancers (Basel), 3(1), 994–1013. doi:10.3390/cancers3010994 [6] Callaway, E. (2015). How elephants avoid cancer. Nature. Retrieved from https://www.nature.com/articles/ nature.2015.18534 [7] American Society of Clinical Oncology. (2020). Li-Fraumeni Syndrome. Retrieved from https://www.cancer.net/cancertypes/li-fraumeni-syndrome [8] Vazquez, J. M., Sulak, M., Chigurupati, S., & Lynch, V. J. (2018). A Zombie LIF Gene in Elephants Is Upregulated by TP53 to Induce Apoptosis in Response to DNA Damage. Cell reports, 24(7), 1765–1776. doi:10.1016/j.celrep.2018.07.042 [9] National Cancer Institute. (2018). Angiogenesis Inhibitors. Retrieved from https://www.cancer.gov/about-cancer/ treatment/types/immunotherapy/angiogenesis-inhibitors-factsheet [10] Nagy, J. D., Victor, E. M., & Cropper, J. H. (2007). Why don't all whales have cancer? A novel hypothesis resolving Peto's paradox. Integrative and Comparative Biology, 47(2), 317– 328. doi:10.1093/icb/icm062 [11] American Cancer Society. (2018). Understanding What Cancer Is: Ancient Times to Present. Retrieved from https:// www.cancer.org/treatment/understanding-your-diagnosis/ history-of-cancer/what-is-cancer.html 基因遺骸LIF6重拾其功能：在 DNA 受損時，LIF6 蛋 白能被 p53 蛋白激活而有效地觸發細胞凋亡，在異常 細胞變成癌細胞前就把它殺死 [8]。這也許解釋了為甚 麼大象可以進化成近蹄類（註二）裡唯一擁有龐大身軀 而相對上不受高癌症風險影響的動物。由此可見，體型 龐大的生物在基因上可能有著獨特的腫瘤抑制機制來 抵禦癌症。 佩托悖論的第二個解釋是腫瘤上的腫瘤 — 超腫瘤 （hypertumors）。癌細胞與正常細胞的不同之處在於 前者喜歡爭奪一切資源，它們並不能與其他細胞和平共 處。腫瘤血管新生（tumor angiogenesis）正能印證 這項特徵，它是指腫瘤生長出為自己提供額外血流的 新血管，藉此奪取額外的氧和營養，好讓自己快速增生 [9]。基於這個事實，我們不難想像腫瘤會用盡一切辦 法霸佔周圍的資源。研究曾預測好競爭的癌細 胞會有利超腫瘤的形成，超腫瘤會寄生在 原本的腫瘤上，藉著吸取母腫瘤血管 內的營養維生 [10]。因此腫瘤 在大型生物體內生長到致 命大小前，超腫瘤可 能早已出現， 透過耗用 母 腫 瘤的資 源使母腫瘤 保持在不會致命的 大小，搗破母腫瘤的邪惡 計劃 [10]。然而，超腫瘤並不會 出現在體型較小的生物中，因為一個 細小腫瘤就已經能威脅到牠們的生命，由 單一癌細胞分裂成致命腫瘤所需的時間相對較 短，因此超腫瘤趕不及形成個體生命就已經危在旦夕。 簡單總結一下，大型生物會出現「癌症的癌症」，從而避 免了癌症發生。 最早關於癌症的描述出自約公元前 3000 年的古 埃及文獻《艾德溫．史密斯紙草文稿》（Edwin Smith Papyrus），當中斬釘截鐵地記述了癌症並 「沒有方法醫治」這個殘酷事實 [11]， 而到了現在亦依然如此，所以稱癌症 為「萬病之王」一點也不為過。探討佩 托悖論的好處在於科學家能從中得知 大自然裡的物種是如何對抗癌症的，以 此為鑑並鑽研新的治療策略。 1 悖論：看似有違常理的矛盾。 2 近蹄類：包含蹄兔和海牛等動物在內的演化支，近蹄類中很多 動物的體型都比大象小（即使是海牛的體型與大象相比還是小 巫見大巫）[8]。
一個都不能遺漏：柵欄錯誤 Not One Less: Fencepost Errors You have f rom 1 p.m. to 4 p.m. to wor k on homework assignments for subjects #1 to #4. If you complete one assignment per hour you should get them all done on time. Yes or no? Now how about if you have from May 1 to May 4 to complete them? Can you get away with completing one assignment per day? These kinds of problems lead to more questions [1]. Why are hours and days counted differently? Where should we start counting from anyway? Applying what we learn in school is never as simple as it seems – even with something as simple as counting. To set things straight, let’s go back to preschool. We learned to count starting from one, two, three … and we also learned that this counting process lets us know how many things there are – pencils, houses, or days. To save time we can just let the labels do the counting for us: The days of the month in June are labelled 1 to 30, so there are 30 days [1]. When we get to subtraction, the teacher holds up six pencils and takes four of them away one by one to demonstrate that 6 – 4 = 2. Subtraction is an arithmetic operation, meaning an action (“operation”) is applied to change the number of objects: In this case, the act of taking a pencil away. Now we are considering a s l ight ly di f ferent concept, the span between two numbers. This isn’t the same as the numbers themselves! Once we are introduced to the number line, we get to represent the subtraction 6 – 4 = 2 as four arrows bumping down from 6 to 2: It’s clear that this refers to the four “spans” between 2 and 6. But the operation actual ly “touches” five numbers: 2, 3, 4, 5, 6. The discrepancy between four and f ive is the fencepost error. Suppose on the way to preschool you pass by an eight-meter fence with posts every two meters [2]. How many fenceposts are there? You might do the mindless division operation and think this means there are four fenceposts in the fence. But in an ordinary fence, with fenceposts at each end, there are actually five. (Is your fence really ordinary? We’ll return to this later.) What went wrong? There are two possible things you can count in this problem: the number of segments of fence between posts, or the number of posts. When By Peace Foo 胡適之
7 you do the division, you start with the total length of the fence (eight meters) and divide by the length of a segment. That will give you an answer of the number of segments of fence. But the question was about the fenceposts, not the fence segments. That means that when dealing with these kinds of problems, the important question is whether you need to count the numbers or the spans [1]. Once you can distinguish between fence segments and fenceposts, it’s quite easy to see the oddities in everyday counting conventions. In music theory, a third denotes an interval of three notes: C-D-E is a third, and so is E-F-G [2]. When you put them together you get C-G, a fifth. In other words, two thirds make a fifth. This is the same as making a longer fence out of two existing fences: You’ll find that there is an extra post left over since the “post” E has been counted twice. The issue is that thirds and fifths refer to the spans between notes but are named for the number of notes they contain: They count the posts instead of the segments [3]. Simi lar ly, fence segments can be disguised as fenceposts. When you celebrate your birthday what you’re actually celebrating is the number of years you were alive [2, 3]. You can even see it in the wording that most people use: On the f i r st bi r thday you celebrate, you turn one year old. The years are the fence segments; the birthdays are the fenceposts. In our original problems, you have four assignments to complete and what you need to do is match them with four time spans in which you can complete them. That means you have to consider the fence segments in the problem. In the wording of the problem, the times 1 p.m., 2 p.m., … , 4 p.m. are markers of time (fenceposts) whi le the days May 1, May 2, … , May 4 are time spans (fence segments) [4]. One gives a span of three hours, the other a span of four days. That’s why your cramming session can fit into one schedule but not the other. A related problem that thi s al so rai ses i s whether your count starts at zero or one [3]. We all know that the first floor can refer either to the ground floor or the floor above it depending on what country you are in. You have ten fingers, but it is possible to count 11 numbers with your fingers: Everyone forgets to include zero fingers [3]. But these are mostly linguistic conventions. More interestingly, think about labeling a series of fence segments #1, #2 … and so on and consider the question of how to label the fenceposts: It naturally requires a fencepost somewhere marked #0. (If you think of the fence segments as timespans spent alive, in years, and the fenceposts as birthdays then this all becomes a version of our birthday discussion above, in which the day of birth can be considered as the zeroth birthday.) This is where many of the problems of confusing fenceposts and fence segments come into play. For this reason, if you reread our or iginal problems about the eight-meter fence, you can see that the question of whether to start counting from zero is tied to whether you want to count fence segments or fenceposts. Counting fence segments is easy. Counting fenceposts requires you to remember that extra zero – zero fingers, the zeroth floor, the zeroth birthday – and add one accordingly [1]. So once again it’s very impor tant to know the context of your quest ion: Should you count the numbers or the spans; fenceposts or fence segments? What kind of fence are you dealing with? Just to leave you with something to think about, suppose the eightmeter preschool fence actually stretches between two walls and doesn’t need posts at either end. Or maybe it closes up to form an enclosure. How many posts are needed now?
你要在下午一時至四時完成科目一至四的功課。如果你 每小時完成一份就能準時完成所有功課，對或錯？ 如果是五月一日至四日呢？如果每天做一份，你能及時 跟功課說再見嗎？ 這些問題背後引伸出更多問題 [1]：為甚麼小時跟日子 的算法會有所不同？我們應怎樣數起？學校教的知識應用 起來永遠不會像想像中簡單 — 顯淺如數算事物也是如此。 要搞清楚這一連串的問題，我們先要回到幼稚園。我 們學會如何從一、二、三……數算物件，亦學會這個數算 過程讓我們知道物件的數量— 不論是鉛筆、房子或是日子。 為了簡化事情，我們為物件標上數字並以此代替數算：六月 的日子被標上數字1 至 30，順理成章地六月就有 30 天 [1]。 然後我們學習減法。老師拿著六枝鉛筆，再把當中四枝 逐枝取走，藉此示範 6 – 4 = 2。減法是算術運算的一種， 指把物件數目改變的動作（「運算」），在上述例子即是取 走鉛筆的動作。現在我們考慮的是稍為不同的概念：兩個數 字之間的間距，這可不是數字本身！在學習數線之後，我們 以由 6 到 2 之間的四個箭頭表示 6 – 4 = 2： 它們清楚表示減法考慮的是由 2 到 6 之間的四個「間 距」，但這次運算其實「碰」到了五個數字：2、3、4、5、6。 四和五之間的分歧就是柵欄錯誤（fencepost error； 註一）了。假設在前往幼稚園的路上你經過一道八米長的籬 笆，當中每兩米豎立了一道籬杆 [2]，那麼籬杆的總數是多 少？你可能不加思索便用除法計算出那裡有四道籬杆，但在 首尾各豎了籬杆的正常籬笆中，籬杆的數目應該是五道。 （你確定那是正常的籬笆嗎？讓我們稍後回到這個問題。） 錯誤出於哪裡？問題中有兩樣東西是你可以數算的：籬 杆之間籬笆的數目，以及籬杆的數目。以除法運算時，你把 籬笆的總長度（八米）除以每段籬笆的長度，那會得出籬笆 有多少段這個數目，但問題問的是籬杆數目而不是籬笆，意 味著解決這些問題時我們必須考慮要數算的是物件還是間 距的數目 [1]。 一旦能分辨出籬笆和籬杆，就很容易發現日常生活中關 於數算的習慣其實充斥著怪誕之處。在樂理中，一個三度是 指相距三個音的音程：C-D-E 是一個三度，E-F-G 也是 [2]， 可是把它們結合時你會得到 C-G，一個五度。換言之，兩個 三度得出一個五度；這就像用兩道籬笆駁成一道更長的籬 笆，數算之下會乍見多出一條「E 柱」。問題的癥結在於三度 和五度理應是指兩個音之間的距離，但卻以當中有多少個 音來命名，即是數算的其實是籬杆而不是籬笆 [3]。 同樣地，籬笆有時也會偽裝成籬杆。慶祝生日時你想紀 念的其實是出生以後活了多少年這個事實 [2, 3]，甚至從日 常用語中也能看出端倪：在你慶祝的第一個生日那時你剛好 一歲。歲數是籬笆，但我們慶祝的是生日，是籬杆。 在原來的問題中，你要完成四份功課，而你要做的是把 四份功課分配在四個時段內完成，所以要考慮的是問題中的 籬笆。在問題的用字上，下午一時、二時……四時等以小時計 的實際上是時間的標記（籬杆），而五月一日、二日……四日 的日子則是時間的間距（籬笆）[4]；前者會給出三小時的間 距，後者則會給出四天的間距。因此你臨急抱佛腳的「趕功 課大計」只能在後者行得通，在前者卻會碰得一臉灰。 這也帶出另一個問題，就是究竟你是由零開始數還是由 一數起 [3]。我們都知道「一樓」在不同國家分別可以指地下 或是地下上面的一層；你有十隻手指，但不少人會忘記如果 比手勢的話你也可以由零的手勢比起，這樣便能數算 11 樣 物件 [3]，但這些大多都只是語言上的習慣。如果你想想，把 一列籬笆逐段標記成 #1、#2……後到底應該如何標記當中 的籬杆，那就更有趣了，因為其中一道籬杆少不免要被標記 為 #0。（如果你把籬笆想成歲數，籬杆想成生日，那其實與 上面生日的例子同出一轍，當中你出生那天正是你的「零歲 生日」。）這就是為甚麼很多關於籬杆和籬笆的問題使人困 惑的根源了。因此，如果你再看一次上面關於八米籬笆的問
9 題，你會發現是否應該由零數起取 決於你想數的是籬笆還是籬杆。數算籬 笆很容易，數算籬杆就需要你在前面加上 那個額外的零 — 零的手勢、零樓（地下或「G 樓」）、零歲生日 — 並在總數加上一 [1]。 因此理解問題的背景非常重要：你要數算的是物件 數目還是間距，籬杆還是籬笆？此外你要處理的籬笆是 哪一種？最後留下一個問題給你想想：假設幼稚園的八 米籬笆兩端原來各自連著牆壁而不需要籬杆，又或是籬 笆圍成一個圓圈而當中沒有缺口，那麼，現在又需要多少 道籬杆呢？ 1 柵欄錯誤：英文「fencepost error」中「fencepost」指的是籬桿，而中 文較常見的翻譯「柵欄錯誤」卻把重點放於籬笆。 References 參考資料： [1] Azad, K. (2009, April 28). Better Explained: Learning How to Count (Avoiding the Fencepost Problem). Retrieved from https://betterexplained.com/articles/learning-howto-count-avoiding-the-fencepost-problem/ [2] Propp, J. (2017, November 16). Mathematical Enchantments: Impaled on a fencepost. Retrieved from https://mathenchant.wordpress.com/2017/11/16/impaledon-a-fencepost/ [3] Parker, M. (2019). Humble Pi: A Comedy of Maths Errors. London, UK: Allen Lane. [4] Lamb, E. (2016, May 10). Roots of Unity: How to Confuse a Traveling Mathematician. Retrieved from https://blogs. scientificamerican.com/roots-of-unity/how-to-confuse-atraveling-mathematician/
The Number Sequence That Here’s a list of numbers: 1, 11, 21, 1211, 111221. Read it to yourself. Now try and guess the next number. Here’s the next number: 312211. And the next one: 13112221. Can you guess the number after that? Th i s puz z l e was g i ven to the ver y famous mathematician John Conway by one of his students. He couldn’t guess it [1]. But the answer is really quite simple: The first number is 1. When you read it to yourself, that’s “one 1”, or 11. Read 11 to yourself: “two 1’s”, or 21. Read 21 to yourself: “one 2, one 1”, or 1211. Read 1211 to yourself: “one 1, one 2, two 1’s”, or 111221. Because it’s generated by reading aloud, Conway called this an audioactive sequence [2, 3], which is also known as a look-and-say sequence. Thi s puz z le apparent l y s tar ted at the 1977 International Mathematical Olympiad [4]. When Conway heard it from a Cambridge math student who had a friend attending the competition, and after he failed to solve it, he decided to make the problem even harder. Why? This is quite standard for mathematicians – when you make a problem harder and more general, it helps you think about how to solve all possible versions of the problem. The obvious way to make the problem harder is to start with any number you like. But one of the first things Conway noticed when starting to work on this problem was that only the digits 1, 2, and 3 “occur naturally” [1]. If you want other digits you’ll need to include them in the first number. So the “interesting” digits are 1 to 3 and we should focus on them if we want to find out more about the problem. More subtly, if you extend your example out far enough you may find something about your problem worth investigating. The next number after 13112221 is 11132 | 13211, then 311312 | 11131221, then 1321131112 | 3113112211. Look at the bars we added that split each number into two parts. If we take just the first part, 11132, and treat it as a single number, we get 311312, then 1321131112 … which are the first parts of the next few numbers. The same thing is true for the second part, 13211. From this point onward the two parts in fact never interact with each other again [1], so Conway called this a “split” and the two parts its “descendants”. He then started looking for numbers that can’t be split in this way. Although there are infinitely many of these numbers, there are exactly 92 of them which must ultimately all appear as the descendants of every possible sequence, except 22, which repeats itself [3]. Since Conway obviously missed studying chemistry in school, he called them “atoms” or “elements”. More complex numbers like 1113213211 that can be split in this way are called “compounds”. So the process of splitting compounds into elements is called “audioactive decay” [5]. Surprise! When do these splits happen? For the string 11132 | 13211, you can see that the first part 11132 ends with 2, so every step from then on ends with “some number of 2’s” and keep ending with 2 no matter what the second part does. On the other side, 13211 begins with a 1 and will continue to begin with either 1 or 3, but not 2, so it will never mess with the first part [1]. Each audioactive element is assigned to one of the first 92 elements of the periodic table, as shown in Figure 1, from hydrogen to uranium. 11132 is hafnium, and 13211 is tin. The names are assigned to resemble the real physical process of radioactive decay into lighter Element Length String The full table can be found on: 92 Uranium 1 3 91 Protactinium 2 13 90 Thorium 4 1113 . . . 1 Hydrogen 2 22 Figure 1 Lengths and strings of some Conway's elements [5]. By Peace Foo 胡適之
11 唸出數列之必要：聽射性數列 Needs to Be Said elements. For example, uranium (3) decays into protactinium (13), then into thorium (1113), and so on. One might expect a lighter element to have a longer string under audioactive decay. However, sometimes an element also decays into a combination of “shorter” elements, which is why some elements are shorter than their predecessors [3] – for instance, promethium (132) fol lows samarium (311332). The next number after 311332 should be 13212312, but it can split into three lighter elements, promethium (132), calcium (12) and zinc (312). In addition, the original starting point of 1, 11, 21, 1211 until 13112221 are referred to as “primordial elements” because they can’t be split either but are not involved in every possible sequence [5]. Conway focused on these 92 numbers, again, because they are “sufficiently general” mathematically; this means they can tell us something interesting about the sequence no matter which starting point we have. It should be possible for you to guess that since al l elements occur in a decay process, any possible decay process wi l l eventually result in only these 92 elements. But as al l mathemat icians know, a guess is not enough. Conway proved this result over about a month of work with assistance from a fel low mathematician and cal led it the “Cosmological Theorem” [3]. Soon later another simpler proof was announced, but unfortunately both proofs were not published. Eventual ly the result was proven again by several others [6]. A consequence of this theorem is that the number of digits of successive numbers increases at a constant rate for all sequences
以下是一個數列：1, 11, 21, 1211, 111221。用中文 （或英文）唸一次，然後猜猜下一個項。 下一個項是：312211。 再下一個是：13112221。 你知道再之後一個項嗎？ 知名數學家 John Conway 的學生曾經叫他猜這道謎 題，但 Conway 沒有成功 [1]。然而答案並不難：第一個項 是 1，對自己唸一次：「一個一」（one 1），那就是 11。唸 一次 11：「兩個一」（two 1’s），所以是 21。唸一次 21：「一 個二，一個一」（one 2 , one 1），1211。唸一次 1211： 「一個一，一個二，兩個一」（one 1 , one 2 , two 1’s）， 111221。由於這個數列是透過朗讀而產生，Conway 稱之 為「audioactive sequence」（像「radioactive」般具「放 射性」，而「audio-」表示與聽力有關）[2, 3]，中文則對應 另一個名字「look-and-say sequence」而常被譯作「外 觀數列」。 這道題目據說源於 1977 年國際數學奧林匹克 [4]。當 Conway 從劍橋數學學生口中得知而一時未能解答後，他 決定把問題改得更難，為甚麼？這其實是數學家解答問題 的標準程序：把問題弄得更困難而更具概括性能有助思考 如何一舉解決問題的所有版本。 使這道問題複雜化的方法明顯就是允許數列從任何數 字開始。Conway 著手解決後其中一樣注意到的是只有數 字 1、2 和 3 是「自然存在」的 [1]；如果想其他數字出現在 數列中，就必須在數列的第一個項加入該數字。因此數字 1 至 3應該內藏著某些「玄機」，如果我們要探究這個問題， 就得專注於這三個數字。 事實上，事情比我們想像的還要巧妙，如果我們把數 列的項都一一寫出，寫到某個位置以後你會發現一些值得 研究的事情。13112221 的下一個項是 11132 | 13211， 然後是 311312 | 11131221，再之後是 1321131112 | 3113112211。留意中間把字串分開的直線，如果我們僅把 前面部分的 11132 取出並當成一個獨立的項的話，之後我 們會得到 311312、1321131112……它們正是原來數列之 後兩個項的前面部分。後面部分的 13211 也是一樣。從這 個項起，字串中的前後部分再不會互相干涉 [1]，Conway 稱這個現象為「分裂」（split）。及後他著手尋找不能分裂 的字串，儘管這樣的字串是無限多的，然而他發現有 92 組 字串既不能分裂，但最終會全數出現在所有可能數列所產 生的項當中（除了22，因為之後所有項都只會是22）[3]。 由於 Conway 實在太掛念在學時讀的化學科，他叫這些字 串「原子」或「元素」，而例如 1113213211 這些稍為複雜、 可以分裂的字串則叫作「化合物」，化合物分裂成元素的過 程就被稱為「聽射性衰變」（audioactive decay）[5]。想 像力夠豐富了吧？ 那甚麼時候會分裂呢？以字串 11132 | 13211 為例， 你可以看見第一部分11132 的結尾數字為 2，不管第二部分 的字串是甚麼，第一部分之後的項都會以 2 結尾。另一方面， 13211 以 1 起首，而之後的項也只能以 1 和 3 起首，絕不可 能是 2，因此它永遠不會與第一部分的字串混在一起 [1]。 每種「聽射性元素」都依照真實元素表上的首 92 種元 素被冠以一個名字，從氫到鈾（見表一）；11132 是鉿，而 13211 是錫。 [3]. In our original sequence starting from 1, these lengths are 1, 2, 2, 4, 6, 6, 8, 10, 14, 20, … which you can calculate for yourself gradually approximates toward a ratio of 1.303577… times the number of digits in the previous term [1]. Based on the proof of the Cosmological Theorem, some application of linear algebra can tell you that this ratio is a solution to some equation of degree at most 92 (i.e. the highest power of the unknown is x92) [3]. Conway and his colleagues then deduced that 1.303577… is actually the largest real root of a specif ic 71-degree polynomial so unnecessarily complex that we’re afraid to print it here [2]. Why does the answer to such a simple sequence involve monstrous decay chains, matr ices and 71-degree equations? We don’t know, but it shows how even the simplest quest ions can produce genuinely engaging mathematics if you know how to look for it. Mathematics is a sport; mathematicians love to challenge themselves. Challenge yourself enough, ask your questions in the right way, and l ike Aladdin’s cave the door to a whole world of interesting new insights will open.
13 References 參考資料： [1] Numberphile. (2014, August 8). Look-and-Say Numbers (feat John Conway) – Numberphile [Video file]. Retrieved from https://www.youtube.com/watch?v=ea7lJkEhytA [2] Bonato, A. (2018, May 2). Audioactive Sequences. Retrieved from https://anthonybonato.com/2018/05/02/ audioactive-sequences/ [3] Conway, J. H. (1987). The Weird and Wonderful Chemistry of Audioactive Decay. In T. M. Cover, & B. Gopinath (Eds.), Open Problems in Communication and Computation (pp. 173–188). New York, NY: Springer. doi:10.1007/978-1-46124808-8_53 [4] Mowbray, M., Pennington, R., & Welbourne, E. (1986). Prelude. Eureka, (46), 4. Retrieved from https://www. archim.org.uk/eureka/archive/Eureka-46.pdf [5] Hilgemeier, M. (1993). Audioactive Decay. Retrieved from http://www.se16.info/mhi/Part1.htm [6] Ekhad, S. B., & Zeilberger, D. (1997). Proof of Conway's Lost Cosmological Theorem. Electronic Research Announcements of the American Mathematical Society, 3, 78–82. 名字分配的方式希望反映真實放射性衰變中元素會衰變成 質量較輕元素的特點，例如鈾（3）衰變成鏷（13），然後是 釷（1113），如此類推。由此我們可能會推論較輕元素應該 在聽射性衰變下擁有較長的字串，但元素有時也會衰變成 由數個較短元素組成的字串組合，令較輕的元素反而比上 一個元素短 [3]，例如接著釤（311332）的是鉕（132）。 311332 的下一個項應該是 13212312，但它可以分裂成 三種較輕的元素：鉕（132）、鈣（12）和鋅（312）。此外， 最初 1、11、21、1211 到 13112221 這數個項被特別命名 為「太初元素」（primordial elements），因為它們不會 衰變，但也不會出現在所有可能的數列中 [5]。重申一點， Conway 只專注於這 92 組字串是因為它們在數學上「足 以概括」（sufficiently general）整條問題；簡而言之，亦 即是無論數列以甚麼字串開首，這 92 組字串都足以告訴 我們一些關於這個數列的有趣事情。 從上文已能猜測出既然全部元素都會出現在一個衰變 過程中，那任何可能的衰變過程最終都只會得出這 92 種元 素。可是，所有數學家都深知「猜測」是不足夠的。Conway 在另一個數學家的幫助下用了一個月時間證明這個猜想， 並稱之為「宇宙論定理」（Cosmological Theorem）[3]。 不久之後出現了更簡單的證明，但不幸地兩份證明都沒有 被公開發表。結果後來被其他數學家重新證明 [6]。 這個定理亦意味著所有數列裡前後兩個項的長度會 以一個固定的率增加 [3]。在我們原來以 1 開始的數列 裡，每個項字串的長度分別為：1、2、2、4、6、6、8、10、 14、20……稍作計算就能發現前後項長度之比會趨近 1.303577…… [1]。參考宇宙論定理的證明，運用一些線 表一 一些 Conway 元素的長度和字串 [5] 元素 長度 字串 全表可見於： 92 鈾 1 3 91鏷 2 13 90 釷 4 1113 . . . 1 氫 2 22 性代數後便會得知這個比例的值是某高次方程（highordered equation）的解，該方程最高次項的次數可達 92 次（即最高次項可為x92）[3]。Conway 和同事之後推 論出 1.303577……其實是某一條 71 次方程的最大實根， 但我們深怕把這條長得可怕的方程列出會嚇怕讀者，故不 在此展示了 [2]。 為何一個如此簡單的數列最終會涉及駭人的衰變鏈、 矩陣和 71 次方程呢？的確很難想像事情為何會向這方向 發展，但它告訴我們只要你懂得怎去尋找，即使是最簡單 的問題也可以衍生出最有趣的數學。數學是一項運動，數學 家喜歡挑戰自己。不時為自己尋求新挑戰，運用智慧問正確 的問題，然後像阿拉丁的奇幻洞窟一樣，一扇通往奇妙新世 界的知識大門就會為你而打開。
By Dana Kim 金娥凜 Let’s start with a riddle. What is the translucent, blue liquid that sits around the bathroom shelves in typical households? You can swish this around your mouth, but you cannot ingest it. Perhaps, you know the answer by now. That’s right – we are referring to a mouthwash! To understand the inspiration behind the invention of modern mouthwash, we need to go back to history, specifically to the 19th century, to catch a glimpse of the development of antisepsis, which refers to the prevention of infections on living tissues by eliminating harmful microorganisms. The British medical scientist, Joseph Lister, was an important figure who paved the way for the development of antisepsis. Unexplained High Surgery Mortality Rates in the 19th Century Back in the 1860s, people were not aware that the culprit of infections was actually microorganisms. Although the idea that microorganisms can cause di sease was establ i shed and promoted by Loui s Pasteur’s famous exper iments around that t ime, medical professionals fai led to associate the germ theory of disease with wound infections. Therefore, medical procedures remained surprisingly primitive by today’s standard. For example, surgical tools were solely cleansed before being placed away for storage. Bed linens were not washed [1] and the same probe was used on the wounds of multiple patients to search for undrained pus [2]. Furthermore, pus was even thought to be a natural component of the healing process – known as “laudable pus” – but they are, in fact, a clear indication of inflammation [3]. As a result, cross infection was very common, especially in surgery, which could lead to gangrene (footnote 1) and deaths [4]. Without knowing the root cause, many surgeons have accepted wound infection to be an inevitable complication of surgery [4]. Some doctors were therefore calling for the abolishment of all surgery, advocating that “the abdomen, chest and brain will forever be closed to operations by a wise and humane surgeon [2].” Introduction of Antiseptic Procedure by Lister This was when Joseph Lister stepped in – inspired by a paper by Pasteur on the use of creosote to disinfect sewage, Lister selected to use carbolic acid (phenol), a newly isolated active ingredient of creosote, to kill germs on the wound and the air above, preventing them from invading the surgical wounds [2, 5]. At first, many scientists were against his idea, as some were nonbelievers of the germ theory [6]. The mainstream belief was that it is the miasma released by the wound itself that causes infections, instead of pathogenic microorganisms, so doctors would wrap the wound with impermeable dressings to insulate the wound from miasma [2, 5]. In addition, some surgeons argued that the antiseptic process would hinder the progress of an operation, in which every second could be crucial to the patient’s survival [4]. However, it was evident that the incorporation of antisepsis prevented major infections from operations, reducing the death rate of Lister’s patients from 46% to 15% [4]. Consequently, the concept of antisepsis became accepted in other countries. Initially, doctors in Germany adopted Lister’s antiseptic procedure, followed by those in the United States, France, and the United Kingdom [2, 7]. It can also be concluded that Lister allowed a broader range of operations to be performed, for instance, abdominal and other intracavity surgery [2]. Inspiration for the Invention of Modern Mouthwash Inspired by the use of carbolic acid for antiseptic surgery, Dr. Joseph Lawrence and pharmacist Jordan Wheat Lambert set out to invent an alcohol-based surgical antiseptic. They eventual ly formulated an antiseptic in 1879 with no specified application [8]. The product failed to take off as a floor cleaner or a cure to dandruff, but became popular after being marketed as an oral care product to prevent plaque and gingivitis [8]. Dr. Lawrence named the modern mouthwash after Joseph Lister – Listeri…a household name you must know! Mouthwashes are now a dai ly necessity sitting around our bathroom shelves, but its development involved at least three prominent scientists in history – Louis Pasteur, Joseph Lister and Joseph Lawrence. So, nex t t ime you r i nse your mouth with mouthwash, 抗菌手術之父：Joseph Lister
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