The notorious Gaokao, (高考, or “High Test”) is China’s SAT on steroids, with a score on the nine-hour test being the sole criterion for admission to Chinese universities. Preparing for the test is a years-long obsession for both students and parents. (In case you missed it: Brook Larmer’s Inside a Chinese Test-Prep Factory.) And for many, the unfortunate consequence is that the lengthy preparation destroys, rather than enhances, academic ability. Student enthusiasm and curiosity are crushed.

声名狼藉的中国高考相当于美国的学术能力评估测试(SAT)，但强度高得多。长达九个小时的高考的成绩是中国大学录取的唯一标准，备考是一件让学生和家长操心多年的事情。（如果你还没读过Brook Larmer的报道“毛坦厂中学，中国应试教育工厂”，请点击此处。）对很多人来说，高考的不幸后果是，学术能力被冗长的备考过程毁掉而不是增强了。学生们的热情和好奇心受到摧残。

Although many in China are aware of the Gaokao’s impact, the test has a 1,300-year history and will not be easily killed. Instead, perhaps the best way to break the Gaokao’s life-draining grasp is indirectly, through clubs and activities that rejuvenate kids’ sense of curiosity and fun. And two weeks ago I discovered one such extra-curricular activity that’s becoming popular among Chinese math-lovers: The Math League, an organization based in New Jersey committed to having kids worldwide enjoy math and discussions about problem-solving.

虽然在中国有很多人知道高考的巨大影响，但这种考试有1300年的历史，并不会被轻易取缔。因此，打破高考令人窒息的控制的最好方法或许是间接的，靠通过各种兴趣小组和活动，重新激发孩子们的好奇心和趣味感。大约两周前，我发现了一项这种课外活动，它正逐渐受到中国数学爱好者的欢迎：数学大联盟杯赛(The Math League)。这家总部设在新泽西的机构致力于让世界各地的孩子喜欢数学，喜欢有关解决问题的讨论。

The Math League organizes math contests around the world, including an annual China vs. United States event held annually at Stanford University. After this competition I had the opportunity to spend a day with the Chinese team introducing the some of the topics (cognitive bias, critical thinking, mathematical creativity, intuitive problem-solving) the students may find useful if they pursue math or computer science as a career. The team was wildly enthusiastic (see photo at the top of this post), perhaps a bit too rowdy at times, but with an abundance of curiosity and self-expression. Any negative impact from the Gaokao seemed, for the moment at least, to be gone.

数学大联盟杯赛在世界各地举办竞赛，包括在斯坦福大学举行年度中美对抗赛。今年的比赛结束后，我有机会和中国队花了一天的时间，向队员介绍一些话题，比如认知偏见、批判性思维、数学创造力，以及直觉性的解决问题方法等。如果学生们想从事数学或计算机科学方面的工作的话，可能会觉得这些话题有用。中国队热情洋溢（见本文顶部的照片），有时可能有点儿过于吵闹，但是他们充满了好奇心，会充分表达自己。任何高考造成的消极影响至少在那个时刻看不到。

Certainly the Math League experience was at the core of this enthusiasm, and this week it’s my pleasure to share the questions used for the competition. (The Math League has copyrighted the questions, and they are used here with permission.) There are quite a few questions, so for our main challenge this week we’ll consider the following five:

参加数学大联盟杯赛的经历当然是这种热情的核心，本周我有幸与大家分享竞赛中使用的试题。（数学大联盟杯赛拥有这些试题的版权，本文经过许可之后使用。）试题很多，因此，作为本周的主要挑战，让我们来考虑下面的五个：

Fourth Annual Stanford Math League Competition (Sample Questions):

第四届斯坦福数学大联盟杯赛年度竞赛（样题）：

1. If my age 24 years ago was 24, what will my age be 24 years from now?

1、如果24年前我是24岁，那么24年后我多大？

2. What is the minimum number of people needed in a room to be certain that at least 3 of them were born in the same month?

2、一间屋子里至少要有多少人才能确保至少有3个人是在同一个月份出生的？

3. If the sum of three prime numbers is 20, what is the largest possible value of one of the three primes?

3、如果三个质数之和为20，那么这三个质数中可能的最大值是多少？

4. If my brain weighs 0.90 kilogram plus half its weight in kilograms, how many kilograms does my brain weigh?

4、如果我的大脑的重量等于0.90千克加上其一半的重量，那么我的大脑究竟有多重？

5. Working alone, a pump fills an empty pool in 3 days. If it works with a second pump, the two pumps together can fill an empty pool in 2 days. In how many days can the second pump, working alone, fill an empty pool?

5、一个水泵独自工作需要3天充满一座空水池。如果加上第二个水泵，两个水泵一起两天就可以充满这座空水池。那么，倘若第二个水泵独自工作的话，需要几天才能充满？

These questions will serve as our official challenge for the week. They aren’t too tough, but how about 60 of them — in 45 minutes? That is the idea behind the Math League’s Speed Test 60. Is the test actually fun? Give it a try with this year’s test.

这些问题是我们本周的正式挑战。它们并不难，但要是有60道题，而且只给45分钟。这就是数学联盟60题快速测试背后的理念。这个测试的确很好玩吧？来试一试今年的试题！

If your pencil’s still sharp and you’re up for more, you might consider the following challenges — also from this year’s Stanford contest. These have tougher questions, but there are only 10 of them. Time limit: 70 minutes for all 10 questions. Here are the Grade 6 and 7 test (10 questions) and the Grade 8 test (10 questions).

如果你还意犹未尽想做更多的题，可以考虑接下来的挑战——也是出自今年的斯坦福竞赛。它包含更难的问题，但是只有10道。此处是六七年级的测试（共10题）和八年级的测试（共10题）。

The average score for Chinese Math League team members, by the way, was 30 out of 60 for the speed test and 5 out of 10 for the individual tests. (Speed Test answers posted at the end of this post. Individual test solutions to be posted on Friday, Sept. 4.)

顺便说一下，中国数学联盟队员的平均成绩是：60道速答题答对30题，个人赛10道试题答对5题。（速答题答案见本文文末。个人赛试题的答案将于9月4日的周五公布。）

I was impressed by the Math League program and asked its co-founder Dan Flegler about its impact. Here’s his response:

数学大联盟杯赛的试题给我留下了深刻的印象。我向这一活动的联合创始人丹·弗雷格勒(Dan Flegler)询问了赛事的影响。下面是他的回答：

I believe that contests such as ours instill a love of mathematics in students and spur them on in their study of mathematics. The best math student I ever taught was getting Cs in the honors ninth-grade math course I was teaching when his father bribed him with a $5 bill to join the math team. On the first contest he ever took, he missed 5 of the 6 questions, but got the very hardest question correct. I was amazed since the problem he got correct was a geometry question that I thought he lacked the requisite knowledge to solve. It turned out that he found a novel way to solve the problem that was completely different from the way I thought it should be done. He went on to complete algebra 1, geometry, algebra 2 and trigonometry in about four months, studying independently under my direction. He eventually won virtually every high school math competition in New Jersey and later got his Ph.D. from Stanford University in applied mathematics. But this never would have happened if he hadn’t been inspired by a math contest.

我觉得，我们举办的这类竞赛，向学生们灌输了对数学的热爱，激励了他们的数学学习。我见过的一名最好的数学学生，他在我教的九年级（相当于中国高一）数学优等班里的成绩只是C，后来他父亲用5美元贿赂他加入了数学竞赛。他首次参赛时，6道题做出了5道，但答对了最难的一道。我很惊讶，因为他做对的那道是几何题，我以为他缺乏必要的解题知识。原来，他发现了一种全新的解题方法，与我认为应该用的方法截然不同。他在我的指导之下，在四个月的时间里，独立完成了代数1、几何、代数2和三角的学习。他后来还在新泽西州的几乎所有高中数学竞赛中获胜，并最终获得斯坦福大学的应用数学博士学位。然而，如果不是因为他的灵感在一次数学竞赛中被激发，这一切也许永远不会发生。

The only students who ever got perfect scores on the speed round were two fraternal Chinese twins (a boy and a girl). Neither I nor any of my colleagues can get a perfect score on the speed round when we take the speed round for fun (even when I write many of the questions myself!). Trying to do 60 questions in 45 minutes isn’t an easy task for anyone. The speed round is really very different from anything else we do in our competitions — we save it for last since it’s like dessert after a heavy meal.

有史以来，只有两名学生做速答题得了满分，他们是来自中国的一对异卵双胞胎（一个男孩和一个女孩）。无论是我、还是我的任何同事，我们为了好玩做速答题时，都得不了满分（尽管其中有不少道题是我自己出的！）。试图在45分钟内做60道题对任何人来说都不是一件容易的事。速答题与我们竞赛中的其他试题完全不一样，我们将速答题保留到最后，因为它们像是一顿大餐后的甜点。

And the Chinese student experience with Math League contests? I asked Rui Hu, coach of the China team (Dr. Hu is at the far right in the photo at the top of this post). Here’s his response:

中国学生参加数学大联盟杯赛的经历如何？我问中国队主教练胡锐博士（本文顶部照片后排最右边的那位）。下面是他的回答：

The Math League and the Stanford International Youth Program enable Chinese students to view and learn math and science from a completely different perspective from what is available in China.

数学大联盟杯赛和斯坦福国际青年计划(Stanford International Youth Program)让中国学生能从另一种角度观察和学习数学及科学，与在中国能得到的角度完全不同。

I find two major problems with current Chinese education. The first is math education. All students are forced to study Math Olympiad problems, very hard problems, even though a lot of these students have no interest in studying these hard problems designed only for math geniuses. This makes Chinese students the least happy students in the world. But Math League contests are well-known math competitions in North America and in the world, with more than 30 years’ history. They present fun and creative problems that promote critical thinking and problem-solving skills, the two most important skills that Chinese students are missing in the current exam-oriented, cram education system in China. It is much more fun working on Math League questions than on extremely hard math problems. Math League contests present a complement to the current Chinese math education. To me, I think the best way is American-style math plus Chinese-style rigorous practice and sound, systematic curriculum.

我觉得目前中国的教育存在两个主要问题。首先是数学教育。所有的学生都被迫学习奥林匹克数学竞赛试题，这些题非常难做，尽管学生中很多人对这些专门为数学天才设计的难题毫无兴趣。这让中国学生成了世界上最不幸福的学生。而数学大联盟杯赛是北美和全球有名的数学竞赛，有30多年的历史。他们出的题既有乐趣，又有创意，有利于提高批判性思维和解决问题的能力，这是在目前中国以考试、应试为主的教育体系中，中国学生所缺少的两个最重要的技能。做数学大联盟杯赛的试题，比做极难的数学题有趣多了。数学大联盟杯赛对目前中国的数学教育是一种补充。在我看来，我认为最好的结合是美国式的数学，加上中国式的严谨练习和完善的系统性课程。

The second problem is English education. Chinese students spend a lot of time on studying English. But Chinese teachers teach students English with the goal to help them achieve high scores in various English tests, such as the TOEFL (Test of English as a Foreign Language) and SAT, not the goal to use English in daily life. So Chinese students achieve very high scores on English tests, but can’t communicate with English speakers. This is true in math education, too. The goal is scoring high in contests like Math Olympiad, not developing students’ true interests in math. So ironically, Chinese students hate math and English, although they spend a lot of time studying math and English. And this is where Math League can help as well. Students can learn native English and have fun solving Math League problems.

第二个是英语教育。中国学生花很多时间学习英语。但中国老师教授学生英语，是以帮助他们在诸如托福（Test of English as a Foreign Language的中文简称）和SAT等各种英语考试上得高分为目标，不是以在日常生活中使用英语为目标。因此，中国学生的英语考试得分很高，但不能与说英语的人交流。在数学教育中也有这个问题。我们的目标是在如奥林匹克数学竞赛上得高分，而不是激发学生对数学学习的真正兴趣。因此有一种讽刺性的后果，那就是中国学生讨厌数学和英语，虽然他们花了很多时间学习数学和英语。而这也是数学大联盟杯赛能提供帮助之处：学生能在学习纯正英语的同时，享受解决数学竞赛试题的乐趣。

And do students focused on the Gaokao really want to take time out for activities like Math League? Here’s Dr. Hu:

集中精力高考的学生真能抽出时间来参加像数学大联盟杯赛这样的活动吗？下面是胡锐的回答：

The current Gaokao system, which was restored in the winter of 1977 by Chinese leader Deng Xiaoping, played the most important and critical role in lifting China from poverty and making China the second-largest economy in the world. Thanks to the Gaokao, millions of quality students are able to enter universities in China; most are trained to be the quality engineers who helped create modern China. But the Gaokao and the education system in both K-12 and university levels have their shortcomings. It is basically a cram system. It doesn’t really encourage critical thinking, creative thinking or intuitive problem-solving. This partially explains why China is very good at copycatting, but not necessarily good at innovation or creativity.

目前的高考制度是中国（已故）领导人邓小平在1977年冬季恢复的，这个制度在让中国脱离贫困、成为世界第二大经济体中起了重要和关键的作用。由于高考，中国数以百万计的优秀学生得以进入大学读书；他们中的大多数都成为中国现代化过程中训练有素的工程师。但高考、以及中小学和大学教育体制也有自己的缺点。基本上是一种应试教育。没有真正鼓励批判性思维、创造性思维，以及直觉地解决问题的能力。这也部分解释了为什么中国能非常好地山寨别人的东西，但创新能力或创造力不一定行。

China has produced many, many quality engineers, but not enough innovators or independent thinkers. The Chinese government realizes this problem, so do Chinese parents and teachers. But it is extremely hard to change the current Gaokao system. No one expects the current Gaokao system will change in foreseeable future. So they (Chinese parents, students and teachers) are looking at alternatives or complementary, and the United States is their first choice, as it is the most powerful and innovative country in the world.

中国已经产生了很多很多训练有素的工程师，但没有足够多的创新者和独立思考者。中国政府意识到了这个问题，中国的家长和老师也意识到了。但是，很难改变目前的高考制度。没有人认为目前的高考制度会在可预见的未来能有改变。因此，他们（中国家长、学生和老师）正在寻找提供替代或补充的东西，美国是他们的第一选择，因为美国是世界上最强大、最有创新力的国家。

Not all the students who participate in Math League contests in China are planning to go to non-Chinese universities. I would say more than half of them are still planning to go to Chinese universities. But these Chinese students want to try something new, something that is innovative and creative, something that is complementary to the current Chinese education system.

并非所有参加数学大联盟杯赛的中国学生打算去国外上大学。我觉得他们中一半以上的人仍打算在中国上大学。但是，这些中国学生想试试新的东西，试一些具有创新性和创造性的东西，一些为目前中国的教育体系提供补充的东西。

Thank you, Dan Flegler and Rui Hu. For more about The Math League, check out its website, which offers information about contests as well as books with past problems.With that final picture of enthusiasm we wrap up this week’s challenge. As always, once you’re able to read comments for this post, use Gary Hewitt’s to correctly view formulas and graphics. (Click here for an intro.) And send your favorite puzzles to gary.antonick@NYTimes.com.

作者感谢丹·弗雷格勒和胡锐。有关数学大联盟杯赛的更多内容见其网站，网站提供竞赛信息以及历届竞赛的试卷。本周的挑战以最后这张热情洋溢的照片结束。与往常一样，如果你能读这篇文章的评论，还可用加里·休威特(Gary Hewitt)提供的工具来正确地显示公式和图形。（点击此处查看工具的有关介绍）。请把你喜欢的智力游戏用电子邮件发给gary.antonick@NYTimes.com。

Solutions

试题答案

The Speed Test brief solutions can be found here. Check back on Friday, Sept. 4, for solutions to the Individual Problems from The Math League.

速答题的简单答案可以在这里找到。请于9月4日的周五到本栏查看数学大联盟杯赛个人赛试题的解答。