📚 Edexcel Decision Mathematics 1 Common Mistakes & Pitfalls | Edexcel 决策数学1 易错点总结
The Edexcel AS and A level Further Mathematics Decision Mathematics 1 module covers algorithms, networks, linear programming, and more. Many students lose marks not because they do not understand the concepts, but because of small, avoidable errors. This article summarises the most common pitfalls found in textbook exercises and e-book resources, helping you to avoid them in the exam.
Edexcel AS 与 A Level 进阶数学决策数学 1 模块涵盖算法、网络、线性规划等内容。许多学生丢分并非由于不理解概念，而是因为一些可避免的小错误。本文总结了教材练习和电子书中常见的易错点，帮助你在考试中避开这些陷阱。
1. Algorithm Efficiency and Order: Understanding Big O Notation | 算法效率与阶：理解大 O 记法
Students frequently confuse the worst‑case orders of sorting algorithms. Bubble sort and shuttle sort are both O(n²) in their worst case, while quick sort is O(n log n) on average. A common error is to claim that an O(n²) algorithm will always be slower than an O(n log n) algorithm; however, for very small values of n the quadratic algorithm can finish first. In an exam question, you must be able to compare orders correctly and state that for large problem sizes, O(n log n) is the more efficient choice.
学生经常混淆各种排序算法的最坏情况阶。冒泡排序和穿梭排序在最坏情况下均为 O(n²)，而快速排序平均为 O(n log n)。一个常见错误是断言 O(n²) 的算法总比 O(n log n) 的算法慢，但实际上当 n 非常小时，二次阶算法可能更快完成。在考试中，你必须能够正确地比较算法的阶，并说明当问题规模较大时，O(n log n) 是更高效的选择。
Another pitfall is ignoring the effect of the initial ordering of the list. Shuttle sort requires only O(n) comparisons when the list is already nearly sorted, whereas bubble sort will still perform multiple passes. Always link the algorithm’s performance to the specific data set described in the question rather than quoting the general worst case without justification.
另一个易错点是忽略列表初始顺序的影响。当列表已经接近有序时，穿梭排序仅需 O(n) 次比较即可完成，而冒泡排序仍需执行多趟扫描。请务必根据题目所给的具体数据集来评价算法表现，而不是不加论证地直接套用通用最坏情况。
2. Kruskal vs Prim: Choosing the Right Algorithm Based on Input | Kruskal 与 Prim：根据输入形式正确选择算法
When a distance matrix is provided, the examiners expect you to use Prim’s algorithm on a table. A common mistake is to extract every edge, list them in order, and then apply Kruskal. This not only wastes time but also increases the risk of missing duplicate edges or making ordering errors. Using Prim on a matrix involves selecting a start vertex, deleting its row, numbering its column, and then repeatedly choosing the smallest uncovered entry from any numbered column.
当题目给出距离矩阵时，考官希望你使用表格形式的 Prim 算法。常见错误是将每条边提取出来、排序，然后套用 Kruskal。这不仅浪费时间，还容易遗漏重边或排错顺序。在矩阵上执行 Prim 算法需要选择起始顶点，删除其所在行并对列进行编号，然后反复从所有已编号列中选出最小的未删除数值。
If a graph is given as a sorted list of edges, Kruskal is usually the quickest choice. Nevertheless, students forget to check for cycles. Always sketch the network as you add each edge and stop when the number of edges equals (number of vertices − 1). An edge that connects two vertices already in the same component must be rejected even if it has the smallest unused weight.
如果图以有序边列表给出，Kruskal 通常是最快捷的选择。然而，学生常常忘记检查是否会形成回路。每次加入一条边时都应随手画出当前的网络草图，当边数达到 (顶点数 − 1) 时即可停止。即使某条边拥有最小的未使用权重，只要它连接了已经属于同一个连通分量的两个顶点，就必须被拒绝。
3. Dijkstra’s Algorithm: Permanent Labels and Working Values | Dijkstra 算法：永久标签与工作值
label. The correct update rule is: new working value = min( current working value, permanent label at the vertex you are leaving + weight of edge ). Always cross out the old working value and write the new one clearly in its place.
Dijkstra 算法中一个极为常见的错误是将边的权重加到了前一个顶点的工作值上，而不是加到其永久标签上。正确的更新规则是：新工作值 = min(当前工作值, 离开顶点的永久标签 + 边权)。请务必划掉旧的工作值，并在原位置清晰地写上新的数值。
Another pitfall concerns breaking ties. When two vertices have the same smallest working value, you may choose either one to make permanent next, but you must record a logical order and keep the remaining working values consistent. Some candidates incorrectly change working values during a tie‑break, leading to a wrong final route. Finally, be sure to write the permanent label inside the box and only stop when the destination vertex’s box has been filled.
另一个易错点涉及如何处理并列最小值。当有两个顶点拥有相同的最小工作值时，你可以任选其一作为下一个永久顶点，但必须记录合理的顺序并保持其余工作值不变。有些考生在解并列时错误地修改了其他工作值，导致最终路线错误。最后，请一定在方框内填写永久标签，并仅在目标顶点的方框已被填好时才停止算法。
4. Critical Path Analysis: Dummy Activities and Float Calculations | 关键路径分析：虚拟活动与浮动时间计算
When constructing an activity‑on‑node network, the most common mistake is inserting unnecessary dummy activities or omitting a dummy that is required to preserve logic. A dummy activity carries zero duration and exists only to show a dependency. Before drawing, list all immediate predecessors; if two activities share some – but not all – predecessors, a dummy may be needed to prevent incorrect dependencies.
在绘制活动节点网络时，最常见的错误是插入不必要的虚拟活动，或者在逻辑关系需要时遗漏虚拟活动。虚拟活动持续时间为零，其唯一作用是表达依赖关系。绘图前应先列出所有活动的紧前活动；若两个活动共享一部分（但非全部）紧前活动，很可能需要添加一个虚拟活动以防止出现错误的依赖。
sh). Free float = (earliest start of the immediately following activity) − (earliest finish of the current activity). A common error is to subtract the current activity’s earliest start from the following activity’s earliest start, which actually gives something closer to the independent float. Always label earliest times and latest times clearly on your diagram and double‑check the forward and backward passes, especially through dummy activities.
浮动时间的各类概念极易混淆。活动的总浮动时间 = 最晚开始时间 − 最早开始时间（或等价地用最晚完成时间 − 最早完成时间）。自由浮动时间 = (紧后活动的最早开始时间) − (本活动的最早完成时间)。常见错误是从后续活动的最早开始时间中减去本活动的最早开始时间，这样算出的结果更接近独立浮动。应在图上清晰标注最早开始/完成时间和最晚开始/完成时间，并仔细检查前后向传递，特别是通过虚拟活动的部分。
5. Linear Programming: Objective Line and Integer Solutions | 线性规划：目标函数线与整数解
A precise objective line is essential. Students often mis‑ awkward points, causing the gradient to be slightly off. This leads to the objective line ‘hitting’ the wrong vertex when slid parallel to the optimal point. Always pick two well‑ uation, and use a ruler. Remember to label the line with its value of P, and clearly indicate the direction of improvement with an arrow or by stating ‘max’/‘min’.
精确的目标函数线至关重要。学生常因选择了不方便的点来绘制直线 ax + by = P，导致斜率略有偏差，从而使目标线在平行移动时“碰到”了错误的顶点。应选取两个相距较远且满足方程的点，并使用直尺绘制。务必在直线上标出对应的 P 值，并用箭头或标注 ‘max’ / ‘min’ 清楚地指出改进方向。
Integer programming questions are a major source of lost marks. After finding the optimal vertex (which may have non‑integer coordinates), you must test all integer points that lie just inside the feasible region near that vertex. Simply rounding the coordinates often gives a point that is either infeasible or not optimal. Draw a small grid or list candidate points and evaluate the objective at each one before stating the final answer.
整数规划问题是丢分的重灾区。在找到最优顶点（其坐标可能非整数）后，必须测试该顶点附近落在可行区域内的所有整数格点。简单地四舍五入坐标往往会产生一个不可行或非最优的点。可以画出小网格或列出候选点，逐一计算目标函数值，再给出最终答案。
6. Hungarian Algorithm: Maximisation and Unbalanced Matrices | 匈牙利算法：最大化与非平衡矩阵
For an assignment problem that asks for a maximum profit or maximum score, you must first convert it into a minimisation problem. Subtract every entry from the largest entry in the matrix. A common error is to swap the roles of minimisation and maximisation partway through the algorithm, forgetting that the final allocation still refers to the original maximisation objective. Always write down the converted matrix and note that the optimal assignment
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