📚 Common Mistakes in Further Maths Decision Maths 1 Exam Questions | 进阶数学决策数学1试卷易错点总结
Decision Maths 1 (D1) is a unique module in Further Mathematics that deals with algorithms, networks, and logical problem-solving. Many students underestimate the precision required in written solutions, leading to repeated errors in exam papers. This article summarises the most frequent pitfalls seen in D1 questions and provides clear guidance on how to avoid them.
决策数学1 (D1) 是进阶数学中一个独特的模块，涉及算法、网络和逻辑问题求解。许多同学低估了笔头解答所需的严谨性，导致试卷中反复出现错误。本文总结了 D1 考题中最常见的易错点，并给出了清晰的避坑指南。
1. Bubble Sort and Quick Sort Efficiency | 冒泡排序与快速排序的效率
Students often confuse the number of comparisons and swaps in bubble sort with the number of passes. In bubble sort, the maximum number of passes is n-1, but the number of comparisons per pass reduces. After each pass, the largest unsorted element is placed correctly, so the number of comparisons in the next pass decreases by one. For quick sort, failing to record the pivot value and clearly showing sublists at each stage costs marks.
同学们常把冒泡排序中的比较次数、交换次数与扫描趟数混淆。冒泡排序的最大趟数为 n-1，但每趟的比较次数是递减的。每完成一趟，当前未排序部分的最大元素就会归位，因此下一趟的比较次数会减少一次。至于快速排序，若没有记录枢轴值并清楚展示每一阶段的子列表，就会失分。
In exam questions, you must state the number of comparisons and swaps explicitly when required. Avoid writing ‘compare first two, swap if needed’ without a structured record; draw a clear table showing the state of the list after each pass or sublist generation.
在考试中，如果要求给出比较和交换次数，必须明确写出。避免仅写“比较前两个，视需要交换”却没有结构化记录；最好用清晰的表格展示每次扫描或每个子列表生成后的列表状态。
2. Binary Search Implementation Errors | 二分查找的实现错误
Binary search mistakes commonly arise from handling the midpoint incorrectly when the list length is even, and from failing to discard the middle term after a comparison. The standard algorithm states: if the middle item is not the target, you discard it and continue searching either the lower or upper half. Some students keep the middle item in the new sublist, which can lead to infinite loops or wrong iterations.
二分查找的常见错误源于列表长度为偶数时中点处理不当，以及在比较后没有丢弃中间项。标准算法规定：若中间项不是目标，则丢弃该项，并在下半区或上半区继续搜索。有些同学会在新的子列表中保留中间项，这可能导致死循环或错误迭代。
Another frequent mistake is miscounting the number of iterations. To be safe, always show a clear table with the left bound, right bound, middle index, and comparison result for each step. When asked for the maximum number of comparisons, use ⌈log₂(n)⌉ or the recurrence relation properly.
另一个常见错误是迭代次数计数错误。为保证准确，应始终用表格列出每一步的左边界、右边界、中点索引和比较结果。当要求给出最大比较次数时，正确使用 ⌈log₂(n)⌉ 或递推关系。
3. Dijkstra’s Algorithm Notation and Working Values | Dijkstra 算法的符号与工作值
The most penalised mistake in Dijkstra’s algorithm is writing final values directly on nodes without showing working values. Examiners expect to see boxes at each vertex, with temporary values replaced by permanent labels in order. The working values must be crossed out and replaced clearly. Also, the order of labelling (permanent labels) must be stated, often in a separate list.
Dijkstra 算法中被扣分最多的错误，是直接在节点上写出最终值而省略工作值。考官期望在每个顶点处画出方框，并依次将临时值划去，替换为永久标号。工作值必须清楚地划掉并更新。此外，标号顺序（永久标号）必须注明，通常可用一个单独列表表示。
Students sometimes forget to record the vertex from which each permanent label is derived. Each time you update a working value, you should note the previous vertex. This is essential for backtracking the shortest path. A common follow-up error is giving a path that does not match the working, or not listing all vertices in the correct sequence for the shortest route.
学生有时会忘记记录每个永久标号来自哪个前驱节点。每次更新工作值时都应注明前一个顶点。这对于回溯最短路径至关重要。常见的后续错误是给出的路径与工作过程不符，或未按正确顺序列出最短路径上的所有顶点。
4. Critical Path Analysis: Dummy Activities and Float | 关键路径分析：虚活动与时差
Identifying the correct number of dummy activities in a precedence table is a persistent challenge. A dummy is needed when two activities share some, but not all, of the same predecessors, or to maintain unique activity-end-node pairings in activity-on-arc representation. Many students add unnecessary dummies or fail to insert a dummy where the logic demands it, altering the network structure.
正确识别优先关系表中虚活动的数量一直是个难点。当两个活动共享部分（而非全部）紧前活动，或者为了在弧表示法中保持活动-结束节点的唯一性时，就需要添加虚活动。很多同学会添加不必要的虚活动，或在逻辑需要时遗漏虚活动，从而改变网络结构。
Float calculations are another minefield. Total float = LST – EST = LFT – EFT. Independent float is often misapplied. The formula for independent float is: EST(next) – EFT(this) – duration. A mistake occurs when students forget to subtract the duration or use the wrong time values. Also, interpreting ‘critical activity’ incorrectly — an activity is critical only if total float is zero (or the minimum possible) and it lies on a critical path.
时差计算是另一个雷区。总时差 = LST – EST = LFT – EFT。独立时差常被误用。独立时差的公式为：EST(后续) – EFT(本活动) – 持续时间。常见错误是忘记减去持续时间，或者使用了错误的时间值。此外，对“关键活动”的误解——只有当总时差为零（或最小可能值）并且该活动位于关键路径上时，它才是关键活动。
5. Linear Programming: Feasible Region and Objective Line | 线性规划：可行域与目标函数线
Graphical linear programming errors frequently involve shading the wrong side of a constraint inequality. A solid convention is to check with the origin (0,0): if the inequality is false at the origin, shade the side not containing the origin. Inconsistent shading leads to an incorrect feasible region marked as R. Another blunder is forgetting that variables must be non-negative, so the axes are constraints too.
线性规划图解法的错误常常与约束不等式取错阴影侧有关。一个可靠的习惯是利用原点 (0,0) 检验：若原点不满足不等式，则阴影在原点异侧。不一致的阴影会导致错误标记可行域 R。另一个疏漏是忘记变量必须非负，因此坐标轴本身也是约束。
When sliding the objective line to find the optimal vertex, students sometimes pick a point that seems optimal but lies slightly outside the feasible region due to rounding. In integer programming problems, using the integer grid within R and testing all possible candidates is vital; guessing the integer solution from a non-integer vertex often loses marks. Always write the coordinates and the corresponding objective value.
在平移目标函数直线求最优顶点时，学生有时会选取一个看似最优但实际上因取整而略超可行域的点。在整数规划问题中，利用 R 内的整数格点并测试所有可能候选点至关重要；根据非整数顶点猜测整数解通常会被扣分。务必写出坐标和对应的目标函数值。
6. Matching: Alternating Path Algorithm and Maximum Matching | 匹配：交错路径算法与最大匹配
The alternating path algorithm is used to improve an initial matching to a maximum matching. A frequent slip is starting the alternating path from an unmatched vertex on the wrong side of the bipartite graph, or not properly alternating between edges not in the matching and edges in the matching. The path must end at an unmatched vertex on the opposite side, otherwise it does not improve the matching.
交错路径算法用于将初始匹配改进为最大匹配。一个常见失误是从二部图错误一侧的未盖顶点开始构造交错路径，或者没有正确地在不属于匹配的边和属于匹配的边之间交替。路径必须在另一侧的未盖顶点结束，否则无法改进匹配。
When writing the alternating path, some candidates omit the sequence of vertices and only state the result, which is insufficient for method marks. Clearly list the path, then flip the status of edges along it. After improvement, the number of matched vertices increases by one. Repeat until no alternating path exists, and state that the matching is maximum.
在写交错路径时，有些考生会省略顶点序列而只给出结果，这不足以获得方法分。应清楚地列出路径，然后翻转沿途边的匹配状态。改进后，已匹配的顶点数增加一个。重复以上步骤直到找不到交错路径，并说明该匹配为最大匹配。
7. Network Flows: Max Flow/Min Cut Consistency | 网络流：最大流与最小割的对应
In flow network problems, the max flow – min cut theorem is often tested. Students must be able to find the maximum flow by augmenting paths and then verify it with a minimum cut. A typical error is stating a cut but not listing the capacities of cut-edges correctly, or miscounting the net flow. The cut must separate the source from the sink, and the capacity of a cut is the sum of capacities of edges going from the source side to the sink side only.
在网络流问题中，常考最大流–最小割定理。学生必须能够通过增广路径找到最大流，然后用最小割加以验证。一个典型错误是给出了割却未正确列出割边的容量，或者算错了净流量。割必须将源点与汇点分开，且割的容量仅包括从源点侧到汇点侧的边的容量之和。
Another mistake is leaving potential backflows unrecorded. When augmenting flow, you must indicate both the additional flow along the forward edge and the reduction in available capacity, as well as the update to the reverse edge’s capacity. Oversimplified diagrams that do not show updated capacities lead to confusion and arithmetic slips.
另一个错误是未记录潜在的回退流量。在增广流量时，必须同时标明沿前向边的附加流量和可用容量的减少，以及反向边容量的更新。简化的图表若未显示更新后的容量，会造成混乱和计算错误。
8. Route Inspection (Chinese Postman) and Odd Vertices | 路径检查（中国邮递员问题）与奇度数顶点
For the route inspection problem, the fundamental step is identifying all odd-degree vertices. Candidates often miss one or mispair them when determining the shortest additional paths. The minimum total extra distance is the sum of the weights of the shortest paths between each pair of odd vertices under an optimal pairing. A common error is not considering all possible pairings and thus not achieving the true minimum.
对于路径检查问题，基础步骤是找出所有奇度数顶点。考生在确定最短附加路径时，经常会遗漏某个奇顶点或配对不当。最小总附加距离等于最优配对下每对奇顶点之间最短路径的权重和。一个常见错误是没有考虑所有可能的配对，从而未能得到真正的最小值。
Once the additional edges are added, the graph becomes Eulerian. The route must start and end at the required vertices (usually the two original odd vertices if the problem asks for a semi-Eulerian traversal, or the same vertex for a full circuit). Forgetting to state the start and end vertices explicitly, or not listing the route clearly, can lose communication marks.
一旦添加了额外边，图就变成了欧拉图。路径必须在要求的顶点处开始和结束（如果题目要求半欧拉遍历，通常是原来的两个奇度顶点；若是完整回路，则起点和终点相同）。忘记明确写出起点和终点，或未清楚列出路径，都会失去表达分。
9. Traveling Salesman Problem: Upper and Lower Bounds | 旅行商问题：上下界
Finding the upper bound using the nearest neighbour algorithm appears straightforward, but small mistakes creep in when choosing the order. You must start at a given vertex, then repeatedly move to the nearest unvisited vertex. If two vertices are equally near, you should state how you break the tie. After visiting all vertices, return to the start. Not adding the return edge to the start is a persistent slip.
用最近邻算法求上界看似简单，但在选择顺序时容易出现小错误。必须从给定顶点出发，反复移动到最近且未访问的顶点。如果两个顶点距离相同，应说明如何打破平局。访问完所有顶点后必须返回起点。忘记添加返回起点的边是一个反复出现的疏忽。
Lower bounds are often more error-prone. The standard method is to find a minimum spanning tree (MST) of the graph and then add the two smallest edges from a chosen vertex to the tree, or by deleting a vertex and finding the MST on the remaining vertices then adding the two smallest connecting edges. Students sometimes use the wrong set of edges or incorrectly apply the shortcut when a vertex has degree constraints. Also, a lower bound can be rejected if it is less than the optimum; examining multiple lower bounds helps to establish the tightest bound.
下界往往更容易出错。标准方法是先找出图的最小生成树（MST），然后将选定顶点的两条最小边加入该树，或者通过删除一个顶点后在剩余顶点上求 MST，再加上两条最小连接边。学生有时会使用错误的边集，或者在顶点度数受限时错误地应用捷径。另外，若某个下界小于最优值，则它可能不够紧；测试多个下界有助于确立最紧的界。
10. Algorithm Tracing and Order Notation | 算法跟踪与复杂度记号
D1 exam papers frequently include tracing an algorithm presented in flow chart or pseudo-code. Misinterpreting a condition box or a loop counter leads to incorrect output. A reliable technique is to construct a trace table with columns for each variable and update it step-by-step. Missing an update or repeating a step because the loop condition was checked incorrectly can alter the entire sequence.
D1 试卷经常要求跟踪以流程图或伪代码给出的算法。误解条件判断框或循环计数器会导致错误输出。一种可靠的方法是构建一个逐列记录各变量的跟踪表格，并逐步更新。遗漏某次更新，或因错误判断循环条件而重复某一步骤，都可能改变整个序列。
Order notation (Big-O) questions assess understanding of algorithm efficiency. Common confusions include saying bubble sort is O(n) in the worst case; it is O(n²) for comparisons and swaps. A common error is using the maximum number of comparisons to argue the order without proper justification. Remember to state the dominant term after removing constants and lower-order terms, e.g. the order of both bubble sort and quick sort worst-case is quadratic, but quick sort best-case is O(n log n) for comparisons.
复杂度记号（大O表示法）的题目考察对算法效率的理解。常见混淆包括认为冒泡排序最坏情况的时间复杂度是 O(n)；实际上比较和交换都是 O(n²)。一个常见错误是利用最大比较次数论证复杂度却没有给出恰当理由。记住要去掉常数和低阶项，指出主导项，例如冒泡排序和快速排序最坏情况均为二次方，但快速排序最好情况的比较次数为 O(n log n)。
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