# 在娱乐问题解决中找到教学法：思考和经验教训

For my PhD I studied at the University of British Columbia (UBC, Vancouver, Canada) where Greg Martin and Dragos Ghioca were organizing a Putnam seminar, which was (is) in part training for the Putnam competition each December. The series was particularly popular with the more ambitious students, whether they planned to take the Putnam or not, and I could see why. No, it wasn’t just because of the free pizza and pop that the department was offering! The problems that were selected for the sessions were (very) difficult and yet deceptively easy to state. To solve them one need not have extensive background beyond the 1st or 2nd year of undergraduate studies, and still they would present an exciting challenge. You had to draw from rather basic material, but to make progress on those problems you would have to use this elementary material in creative ways. I always found this challenge very rewarding, and at UBC it was clear that I wasn’t the only one who felt this way.

My current problem solving sessions (PSS) are held every two weeks in both semesters. The sessions last 1.5-2 hours. They take place in a big lecture hall (when in person) or Blackboard–our Virtual Learning Environment—while all teaching and learning activities were moved online due to the pandemic. When in person, there is always pizza (which is a bit counterintuitive, I am always feeling “slower” after a couple of slices!) and very little structure. I distribute the handouts and then people move around, use the boards, chat, brainstorm, sigh loudly in frustration, but also exclaim in delight when a breakthrough comes. I love it! There are 110+ members in my mailing list for the group and we typically see 30-70 students and faculty joining the sessions.;我estimate the regular members around 40, the vast majority of whom are undergraduate students.

$[n = d_k 10^k + d_ {k-1} 10^{k-1} + \ cdots + d_0 10^0]$，

$d_k \ neq 0$ 和$d_i \ in {0,1,2，\ dots，10}$ for All $i$。

For instance, the integer $N = 10$ has two base 10 over-expansions: $10 = 10 \cdot 10^0$

Can we be more systematic in our search? Maybe, let’s look at 2-digit numbers first, then 3-digit numbers, and so on?

List your findings. (we are trying to spot a pattern!)

1-9：独特的过度膨胀

10-99：如果倍数为10，即形式 *0。嗯……不是唯一的。

100-999: not unique if multiple of 10, i.e., of the form * * 0 , but wait! I can also find another representation for numbers of the form * 0 * . Hmm…

1000-9099：让我们看看，我可以找到与表单 * * * 0， * * 0 *， * 0 * *的标准基本10表示形式不同的。嗯...！

One of the hardest aspects in organizing this activity was striking a good balance: and trying to be more inclusive, trying to adjust the level of the problems so that students don’t get totally discouraged, but at the same time making them hard enough for students to think outside the box, embrace new techniques and the “investigation-first” attitude that I am hoping to promote; and to also build resilience in the face of challenge. I don’t claim to have found an algorithm for selecting good problems, and I am sure some keen students have been occasionally discouraged despite my best intentions. Other than managing expectations and working towards a supportive environment where “failing” is seen as a necessary (and expected) step of the process, I have tried to adapt too: I have been starting the sessions off with quick, “warm-up” problems, still not trivial, that everyone has a chance to solve. (Groups of) students work independently, so nobody will notice if you spend the whole session working on the “warm-up problem”. Note to self: stop calling it a “warm-up” problem?

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