陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「New version of algebra game」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Just a quick update on my previous post on gamifying the problem-solving process in high school algebra. I had a little time to spare (on an airplane flight, of all things), so I decided to rework the mockup version of the algebra game into something a bit more structured, namely as 12 progressively difficult levels of solving a linear equation in one unknown . (Requires either Java or Flash.) Somewhat to my surprise, I found that one could create fairly challenging puzzles o

After completing each level, an icon appears which one can click on to proceed to the next level. (There is no particular rationale, by the way, behind my choice of icons; these are basically selected arbitrarily from the default collection of icons (or more precisely, “costumes”) available in Scratch.)

已知结果和反例

The restriction of moves made the puzzles significantly more artificial in nature, but I think that this may end up ultimately being a good thing, as to solve some of the harder puzzles one is forced to really start thinking about how the process of solving for an unknown actually works. (One could imagine that if one decided to make a fully fledged game out of this, one could have several modes of play, ranging from a puzzle mode in which one solves some carefully constructe

One advantage to gamifying linear algebra, as opposed to other types of algebra, is that there is no need for disjunction (i.e. splitting into cases). In contrast, if one has to solve a problem which involves at least one quadratic equation, then at some point one may be forced to divide the analysis into two disjoint cases, depending on which branch of a square root one is taking. I am not sure how to gamify this sort of branching in a civilised manner, and would be interest

证明或构造的主线

先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。

阅读时建议盯住的点

先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源