📚 Further Maths Decision Maths 1 Common Mistakes | 决策数学1 常见易错点总结
Decision Maths 1 is a fascinating branch of Further Mathematics that focuses on algorithms, optimization and logical problem-solving. However, students often lose marks not because they do not understand the concepts, but because of subtle mistakes in applying algorithms, interpreting results or handling special cases. This article summarises the most common pitfalls across the key topics of Decision Maths 1, including sorting and searching, graph theory, minimum spanning trees, bin packing, linear programming, the simplex algorithm, critical path analysis and matchings. Every section pairs the typical error with the correct procedure, so you can turn a mark-losing habit into a mark-winning routine.
决策数学1是进阶数学中一个迷人的分支，专注于算法、优化和逻辑问题求解。然而，学生丢分往往不是因为不理解概念，而是因为在应用算法、解读结果或处理特殊情况时出现细微错误。本文总结了决策数学1各核心主题中最常见的易错点，涵盖排序与搜索、图论、最小生成树、装箱问题、线性规划、单纯形法、关键路径分析和匹配等内容。每一节都把典型错误与正确做法配对讲解，帮助你把丢分的习惯变成得分的习惯。
1. Sorting and Searching: Algorithms: Off-by-One and Termination Mistakes | 排序与搜索：差一错误与终止条件失误
A classic bubble sort error is failing to reduce the number of comparisons after each pass. After the first pass the largest item is already in its final position, so the second pass should compare only the first n − 1 items, the third pass n − 2 items, and so on. Continuing to compare the whole list wastes time and loses the "efficiency" mark. Remember that a list of n items needs at most n − 1 passes.
冒泡排序最典型的错误，是在每一轮之后没有减少比较次数。第一轮结束后，最大的元素已经就位，所以第二轮只需比较前 n − 1 个元素，第三轮比较前 n − 2 个，以此类推。继续比较整个列表既浪费时间，也会丢掉"效率"分。要牢记：含 n 个元素的列表最多需要 n − 1 轮。
Comparisons in bubble sort = (n − 1) + (n − 2) + … + 1 = n(n − 1) ÷ 2
point or write L = mid instead of L = mid + 1, which causes an infinite loop. The search terminates when L > R, and you must state the number of comparisons actually used.
在二分搜索中，列表必须已经排好序，且中点必须用向下取整公式求出。学生常常更新错误的端点，或把 L = mid + 1 误写成 L = mid，结果造成死循环。搜索在 L > R 时终止，而且你必须写出实际使用的比较次数。
mid = ⌊(L + R) ÷ 2⌋ , stop when L > R
In quick sort, after partitioning, the pivot is in its final position and must be excluded from both sublists. In insertion (shuttle) sort, the mistake is comparing only with the immediately preceding item instead of working backwards through the sorted section. Whatever the algorithm, always show the intermediate lists after each pass, because method marks are awarded for the working, not just the final ordered list.
在快速排序中，划分之后主元已经处于最终位置，必须把它排除在两个子列表之外。在插入排序（穿梭排序）中，错误在于只与前一个元素比较，而不是沿着已排序部分向前依次比较。无论使用哪种算法，都要写出每一轮之后的中间列表，因为方法分是给过程的，而不仅仅是给最终有序列表的。
2. Flow Charts and Trace Tables | 流程图与追踪表
Flow chart symbols are examined directly, so you must know them precisely: a rectangle is a process or instruction, a parallelogram is input or output, a diamond is a decision, and rounded terminals mark the start and the end. A very common error is writing a condition inside a process box, or drawing a decision with only one exit arrow.
流程图的符号是直接考查的内容，因此必须准确掌握：矩形表示处理或指令，平行四边形表示输入或输出，菱形表示判断，圆角终端表示开始和结束。最常见的错误是把判断条件写进处理框里，或者画出的菱形只有一个出口箭头。
When filling in a trace table, record the value of every variable after every step, not only at the points where the value changes dramatically. A frequent mistake is to skip the final iteration, or to record the output before the loop condition has been tested one last time. Read the algorithm literally, line by line, and never assume what the programmer "meant".
填写追踪表时，要记录每一步之后每个变量的值，而不仅仅是在数值大幅变化的地方记录。常见的错误是跳过最后一次循环，或者在循环条件最后一次判断之前就写出输出结果。要逐行照着算法字面执行，绝不要臆测"作者本意"。
Another subtle trap is confusing assignment with comparison. In pseudocode, "SET x = x + 1" means replace the value, whereas "IF x = 5" rns a correct algorithm into a nonsense one and costs both accuracy and explanation marks.
另一个隐蔽的陷阱是把赋值与比较混淆。在伪代码中，"SET x = x + 1"表示替换数值，而"IF x = 5"是判断。把两者混用会把正确的算法变成无意义的算法，既丢准确性分，也丢解释分。
3. Dijkstra's Algorithm: Labels, Order and Route Reconstruction | 迪杰斯特拉算法：标签、顺序与路径回溯
Dijkstra's algorithm is the highest-frequency topic in Decision Maths 1, and almost every lost mark comes from the labelling convention. Each vertex should carry a box with three entries: the order of labelling, the final (permanent) distance from the start, and the preceding vertex. The start vertex is always order 1 with distance 0.
迪杰斯特拉算法是决策数学1中出现频率最高的考点，而几乎每一处失分都来自标签的书写规范。每个顶点应带有一个包含三项内容的方框：标号顺序、距起点的最终（永久）距离、以及前驱顶点。起点永远是顺序 1、距离 0。
A very common error is to label a vertex permanently before checking whether a shorter route has just become available. Every time a new vertex becomes permanent, you must check all its unlabelled neighbours and update any temporary label that can be improved. Only then choose the smallest temporary label as the next permanent one.
最常见的错误是在还没有检查是否出现更短路径之前，就把某个顶点标为永久。每当一个新顶点成为永久标号，你必须检查它所有未标号的相邻顶点，并更新任何可以改进的临时标号。之后才能选择最小的临时标号作为下一个永久标号。
When reconstructing the shortest path, work backwards from the destination using the third entry of each box, and always state the total length with its units. On a directed graph, arcs may only be travelled in the direction of the arrow, and a route that uses an arc backwards is automatically wrong.
回溯最短路径时，要利用每个方框的第三项，从终点倒着推回起点，并且一定要写出总长度及其单位。在有向图中，弧只能沿箭头方向通行，任何逆向使用弧的路线都是错的。
Finally, remember the scope of the algorithm: Dijkstra fails if any arc has a negative weight. If a question includes a negative weight, the correct response is to state that the algorithm is not valid here, not to carry on regardless.
最后要记住算法的适用范围：只要存在负权弧，迪杰斯特拉算法就会失效。如果题目中出现负权，正确做法是指出该算法在此不适用，而不是硬算下去。
4. Kruskal's and Prim's Algorithms: Minimum Spanning Trees | 克鲁斯卡尔与普里姆算法：最小生成树
For Kruskal's algorithm, list all arcs in ascending order of weight, then add an arc only if it does not form a cycle with the arcs already chosen. The single most common error is carrying on after n − 1 arcs have been selected for a graph with n vertices, or adding an arc that closes a loop. Stop as soon as you have n − 1 arcs.
使用克鲁斯卡尔算法时，先把所有弧按权值从小到大排列，然后只在不与已选弧构成回路时才加入该弧。最常见的错误是：对于含 n 个顶点的图，在已经选出 n − 1 条弧之后还继续选，或者加入了会形成回路的弧。一旦得到 n − 1 条弧就应立即停止。
Number of arcs in a spanning tree = n − 1 (n vertices, no cycles)
For Prim's algorithm, start from the vertex specified in the question and build a table of the nearest connected and unconnected vertices. The classic mistake is failing to update the table after each new vertex joins the tree: when a vertex is added, all arcs from that vertex to the remaining unconnected vertices must be compared and the smaller value recorded.
使用普里姆算法时，从题目指定的顶点出发，建立"已连接顶点—最近未连接顶点"的表格。典型错误是在每次有新顶点加入树之后没有更新表格：当某顶点被加入时，必须把它到所有剩余未连接顶点的弧进行比较，并记录较小的权值。
Both algorithms produce a minimum spanning tree with the same total weight, so a useful check is to run the second algorithm and confirm the total. Always write the final total weight with units, and if the question says "state the arcs in the order in which they are chosen", the order itself carries marks.
两种算法得到的最小生成树总权值相同，因此一个有用的检验方法是用另一种算法再算一遍并核对总权值。一定要写出最终总权值及其单位；如果题目要求"按选择顺序写出各条弧"，那么顺序本身就是得分点。
5. Bin Packing and the Packing Heuristics | 装箱问题与装箱启发式算法
First-fit requires you to scan the bins from the very beginning every single time, placing the item in the first bin with enough remaining space. Students often continue from the last bin used, which turns first-fit into next-fit and gives a different (and wrong) answer. First-fit decreasing requires the items to be sorted into descending order first, otherwise the answer is simply not FFD.
首次适应法要求每一次都从第一个箱子开始扫描，把物品放入第一个剩余空间足够的箱子。学生常常从上次用过的箱子继续向后找，这就把首次适应法变成了下次适应法，得到的结果自然不同（而且是错的）。首次适应递减法要求先按物品大小降序排列，否则答案就不是首次适应递减法的结果。
A useful lower bound on the number of bins is the ceiling of the total size divided by the bin capacity. Never claim that a heuristic has found the optimal packing unless you have justified it with such a bound and shown that the packing achieves it.
箱子数量的一个有用下界是：物品总体积除以箱子容量，再向上取整。除非你用这个下界加以论证并说明该装箱方案达到了这个下界，否则绝不能声称某个启发式算法找到了最优解。
Lower bound = ⌈ (Σ item sizes) ÷ bin capacity ⌉
In full-bin packing, the mistake is to assume that a perfect combination always exists. If no combination exactly fills a bin, you must state that full-bin packing cannot be applied and fall back on another heuristic, explaining your reasoning in one clear sentence.
在满箱法中，错误在于假定总能找到恰好装满的组合。如果不存在恰好填满一个箱子的组合，你必须说明满箱法无法使用，并改用其他启发式算法，用一句清晰的话解释理由。
6. Graph Theory Terminology and Traversability | 图论术语与可遍历性
Precise vocabulary earns easy marks, and imprecise vocabulary loses them. A loop contributes 2 to the degree of a vertex, and parallel edges each contribute 1. The handshaking lemma states that the sum of all degrees equals twice the number of edges, which is the quickest way to check a degree table.
准确的术语能轻松得分，含糊的术语则直接丢分。一个环对该顶点度数的贡献是 2，而平行边每条各贡献 1。握手引理指出，所有顶点的度数之和等于边数的两倍，这是检验度数表最快的方法。
Σ degrees = 2 × (number of edges) , edges in Kₙ = n(n − 1) ÷ 2
Traversability rules must be quoted exactly. A connected graph with every vertex of even degree has an Eulerian circuit and can be traversed starting and finishing at the same vertex. A connected graph with exactly two odd vertices is semi-Eulerian: the trail must start at one odd vertex and finish at the other. With four or more odd vertices, no single trail covers every edge.
可遍历性判据必须原样引用。若连通图的所有顶点度数均为偶数，则存在欧拉回路，可以从同一顶点出发并回到该顶点走遍所有边。若连通图恰有两个奇度顶点，则是半欧拉图：路径必须从一个奇度顶点出发，在另一个奇度顶点结束。若有四个或更多奇度顶点，则不存在一条走遍所有边的路径。
The most damaging confusion is between Eulerian and Hamiltonian. An Eulerian trail uses every edge exactly once; a Hamiltonian cycle visits every vertex exactly once. Questions about "traversing every road" are Eulerian, while questions about "visiting every town" are Hamiltonian.
最有害的混淆是把欧拉与哈密顿弄反。欧拉路径恰好经过每条边一次；哈密顿回路恰好访问每个顶点一次。凡是"走遍每条道路"的问题属于欧拉类，而"访问每个城镇"的问题属于哈密顿类。
7. Linear Programming: Formulating the Model | 线性规划：建立模型
Formulation errors are the most expensive mistakes in this topic because they destroy every later mark. Always start by defining the variables in words, including the units, for example "let x be the number of small crates produced per day". Then write the objective function, then the constraints, then the non-negativity conditions.
建模错误是这一考点中最"昂贵"的失误，因为它会连带毁掉后面所有的分。永远先用文字定义变量并写明单位，例如"设 x 为每天生产的小箱数量"。然后写目标函数，再写约束条件，最后写非负条件。
The direction of the inequality is where most marks vanish. Phrases such as "at least", "no fewer than" and "minimum of" give ≥, while "at most", "no more than" and "maximum of" give ≤. A phrase such as "the number of large crates is at least twice the number of small crates" becomes y ≥ 2x, not x ≥ 2y.
不等号方向是失分最集中的地方。"至少""不少于""最少"对应 ≥，而"至多""不超过""最多"对应 ≤。例如"大箱数量至少是小箱数量的两倍"应写成 y ≥ 2x，而不是 x ≥ 2y。
When reading the optimal solution from a graph, evaluate the objective function at every vertex of the feasible region, or use a ruler to slide the objective line parallel to itself. A frequent error is to test only the vertices that look extreme, or to forget that the optimum may lie at a non-integer vertex when integer answers are not required.
从图像读取最优解时，要在可行域的每个顶点上计算目标函数值，或者用直尺平移目标函数线。常见错误是只检验看起来最"极端"的顶点，或者忘记在题目不要求整数解时，最优点可能落在非整数顶点上。
8. The Simplex Algorithm: Tableaux, Pivots and Slack Variables | 单纯形法：表格、主元与松弛变量
needs its own slack variable, and the objective must be rewritten so that all variables appear on the left with the constant on the right. Losing a slack variable makes the entire tableau meaningless.
第一个错误通常出现在任何运算之前。每个 ≤ 约束都需要一个自己的松弛变量，目标函数也必须改写，使所有变量出现在左侧、常数在右侧。漏掉一个松弛变量会让整张表格失去意义。
The pivot is chosen in the column with the most negative entry in the objective row, and the pivot row is chosen by the ratio test: divide each positive entry in the pivot column into the corresponding value in the "value" column, and pick the smallest non-negative ratio. Using a negative or zero entry in the ratio test is the classic error, because such a ratio is invalid and must be ignored.
主元所在列由目标行中负得最多的元素确定，主元所在行则由比值检验确定：把主元列中每个正元素分别去除"数值"列中对应的数值，取最小的非负比值。用负数或零做比值检验是典型错误，因为这种比值无意义，必须忽略。
After choosing a pivot, clear the pivot column completely, including the objective row, using row operations. The common slips are forgetting to divide the pivot row itself by the pivot, and arithmetic errors in the third or fourth iteration. Always write the full tableau at each stage: method marks are given for visible correct row operations.
选定主元之后，要用行变换把主元列(包括目标行)全部化为 0，只留主元为 1。常见疏漏是忘记先用主元去除主元行本身，以及在第三、第四次迭代中出现运算错误。每一阶段都要完整写出表格：方法分是给可见的正确行变换的。
Finally, read the answer from the final tableau with care. The value of the objective is the entry in the value column of the objective row, and each basic variable equals the value in its own row. Do not round intermediate tableaux, and do not forget to state the optimal values of all the original variables, not just the objective.
最后，要从最终表格中仔细读出答案。目标函数的取值就是目标行"数值"列中的数，每个基变量等于它所在行中的数值。中间表格不要提前四舍五入，也不要只写目标值而忘记给出所有原始变量的最优值。
9. Critical Path Analysis: Activity Networks and Dummy Activities | 关键路径分析：活动网络与虚活动
In an activity-on-arc network, each arc represents an activity and each node is an event. A dummy activity has duration zero and exists for two reasons only: to preserve a precedence relationship that the arcs alone cannot express, and to keep two activities with the same start and end events distinct. Adding a dummy that is not needed is just as wrong as omitting one that is.
在"弧表示活动"的网络中，每条弧代表一项活动，每个节点是一个事件。虚活动的持续时间为零，其存在只有两个理由：一是表达仅靠弧无法表达的先后关系，二是把起止事件相同的两项活动区分开来。添加不必要的虚活动，与漏掉必要的虚活动一样属于错误。
The forward pass gives the earliest start time of each activity, and the backward pass gives the latest finish time. The trap is to exchange the two: EST is calculated from the start forwards using the maximum, while LFT is calculated from the end backwards using the minimum. Mixing them up produces negative floats, which should be a warning sign that something is wrong.
前向遍历给出每项活动的最早开始时间，后向遍历给出最迟完成时间。陷阱在于把两者弄反：EST 是从起点向前用最大值计算，而 LFT 是从终点向后用最小值计算。把它们弄反会产生负的时差，这本身就是"出错了"的警示信号。
Always write the earliest and latest event times in the correct halves of the node box, and name the critical path by listing the activities in order. The critical path is the longest route through the network, and its length equals the minimum project completion time.
务必把事件的最早时间与最迟时间写在节点框中正确的位置，并按顺序列出活动名称来写出关键路径。关键路径是网络中耗时最长的路线，其长度等于项目的最短完成时间。
10. Floats, Gantt Charts and Resource Levelling | 时差、甘特图与资源平衡
Total float is the amount of time an activity can be delayed without delaying the whole project. It is calculated from the latest finish time, the earliest start time and the duration. Critical activities have zero total float; an activity with a float of zero is not necessarily on the critical path if it lies on a parallel route of equal length, so check the network rather than the number alone.
总时差是指一项活动在不延误整个项目的前提下可以推迟的时间，由最迟完成时间、最早开始时间和持续时间共同决定。关键活动的总时差为零；但总时差为零的活动不一定就在关键路径上，如果它位于另一条等长的平行路线上，就需要回到网络图中核对，而不能只看数字。
Total float = LFT − EST − duration
When drawing a Gantt or cascade chart, the vertical axis often shows the number of workers and the horizontal axis shows time. The common errors are drawing an activity before its predecessors have finished, overrunning the stated worker limit at some point in time, and forgetting that resource levelling may extend the project duration, whereas resource smoothing tries not to.
绘制甘特图或瀑布图时，纵轴通常表示工人数量，横轴表示时间。常见错误包括：在紧前活动尚未完成时就画出某项活动；在某个时间点上超出了题目规定的工人数量上限；以及忘记资源平衡可能会延长项目工期，而资源平滑则力求不延长。
Read the wording of the question carefully. "Delay the start of activity G as far as possible" means use the whole float, while "state the float of activity G" simply wants the number. Candidates who confuse the two lose an easy mark every session.
要仔细读题。"尽可能推迟活动 G 的开始时间"意味着用完全部时差，而"写出活动 G 的时差"只是要一个数值。每一考季都有考生把两者混淆，白白丢掉一个容易的分。
11. Matchings and the Augmenting Path Algorithm | 匹配与增广路径算法
A matching pairs vertices from one set with vertices in another set so that no vertex is used twice. A maximum matching uses as many pairs as possible, while a complete matching requires every vertex on one side to be matched. The two are not the same thing, and questions often test the distinction directly.
匹配是把一个集合中的顶点与另一个集合中的顶点配对，且任何顶点都不能被重复使用。最大匹配使用尽可能多的配对，而完全匹配要求一侧的每个顶点都被匹配上。两者并不相同，题目常常直接考查这一区别。
In the augmenting path algorithm, start from an unmatched vertex on the left, then alternate between arcs not in the matching and arcs in
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图