Learning Objectives 學習目標
4 Goals 四大目標
Q: Name 3 control structures.
說出三種控制結構。
Key Term 關鍵詞
Nested Structure
嵌套結構
將一個結構放入另一個結構
Q: What is nested structure?
嵌套結構是什麼？
Key Term 關鍵詞
Nested Conditional
Statement 嵌套條件語句
根據前一條件的結果決定是否進入另一條件語句
Key Term 關鍵詞
3 Control Structures
三種控制結構
順序 / 選擇 / 迭代
Q: Which structure repeats?
哪種結構重複執行？
Tip 提示
Nested = placed
inside another
嵌套 = 放入另
一個結構之內
Application 應用
→ 2D arrays + nested loops
= image manipulation!
→ 二維陣列 + 嵌套循環
= 圖像處理！
Chapter 3 Learning Objectives 學習目標
- Understand how to use nested control structures
了解如何使用嵌套控制結構
- Integrate and apply various control structures to design solutions
綜合運用各種控制結構設計解決方案
- Evaluate algorithms by efficiency, correctness & appropriateness
按效率、正確性和適切性評估演算法
- Select appropriate algorithm based on complexity, data structures & trade-off
根據複雜性、數據結構及取捨選擇適當演算法
3.1 Nested Control Structures 嵌套控制結構
Placing a structure inside another = nested structure 嵌套結構.
將一個結構放入另一個結構之內，稱為嵌套結構。
A nested conditional statement 嵌套條件語句 decides whether to enter another conditional statement based on the result of the previous condition.
嵌套條件語句根據前一條件的結果，決定是否進入另一條件語句。
Two Forms of Nested if 嵌套 if 的兩種形式
| Form 形式 |
Python Code |
Notes 備注 |
if-elif chain
鏈式 elif |
if (c1):
body1
elif (c2):
body2
else:
body3 |
Conditions checked
in sequence 按順序
檢查條件 |
Truly nested
真正嵌套 |
if (c1):
if (c2):
body2
else:
if (c3):
body3 |
Inner if inside
outer branch
內部 if 在外部
分支之內 |
Key: A conditional can be nested in the else branch OR any other branch.
條件語句可嵌套在 else 分支或任何其他分支中。
Triangle Example 三角形例子 (Checkpoint 3.1)
Input: 3 side lengths a, b, c → Classify triangle type:
輸入三條邊長，判斷三角形類別：
- Equilateral 等邊三角形 — all 3 sides equal (三邊相等)
- Isosceles 等腰三角形 — exactly 2 sides equal (兩邊相等)
- Scalene 不等邊三角形 — all sides different (三邊不等)
- Invalid 不能構成三角形 — any 2 sides sum ≤ third side
Summary 總結 (Page 1)
Nested structure = placing one structure inside another (嵌套結構 = 將一結構放入另一結構).
Nested conditional: inner if runs based on outer condition result (內部 if 根據外部條件結果執行).
Two forms: elif chain (鏈式) vs truly nested (真正嵌套). Indentation is critical! (縮排非常重要)
3 control structures:
Sequence 順序
Selection 選擇
Iteration 迭代
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Key Term 關鍵詞
Nested Loop 嵌套循環
將一個循環放入另一個循環
Q: What is outer/inner loop?
什麼是外/內循環？
Key Term 關鍵詞
Outer Loop 外循環
控制內循環執行的次數
Key Term 關鍵詞
Inner Loop 內循環
每次外循環執行時，
內循環完整執行一遍
Q: If outer=2, inner=3,
how many total?
總共執行多少次？
Formula 公式
Total inner reps
總內循環次數
= outer reps
× inner reps
外循環次數 × 內循環次數
Analogy 類比
Clock 時鐘:
Minute hand = outer loop
Second hand = inner loop
分針 = 外循環
秒針 = 內循環
3.2 Nested Loop 嵌套循環
Placing a loop inside another loop = nested loop 嵌套循環.
將一個循環放入另一個循環之內，稱為嵌套循環。
Pseudocode
Outer loop 外循環
for i from 0 to 1
Output i
Inner loop 內循環
for j from 0 to 2
Output j
Python
for i in range(0, 2):
print(i)
for j in range(0, 3):
print(j)
Trace Table for Nested Loop 嵌套循環追蹤表
| Line 行 | i | j | Output 輸出 | Note 備注 |
|---|
| for i ← 0 | 0 | | | Outer body
(i=0)
外循環體 |
| Output i | | | 0 |
| for j ← 0 | | 0 | 0 | |
| for j ← 1,2 | | 1,2 | 1, 2 | |
| for i ← 1 | 1 | | 1 | Outer body
(i=1) |
| for j ← 0,1,2 | | 0,1,2 | 0, 1, 2 | |
Each time outer loop body executes → inner loop body executes 3 times.
每次外循環體執行 → 內循環體執行 3 次。
Total inner repetitions = 2 × 3 = 6
Counting Repetitions 計算重複次數
| Code | Outer reps 外 | Inner reps 內 | Total 總計 |
|---|
| for i 0→1, for j 0→2 | 2 | 3 | 6 |
| for i 1→5, for j 1→6 | 4 | 5 | 20 |
| range(0,5), range(0,10) | 5 | 10 | 50 |
Example: Draw a Square of * (Example 3.1)
L = int(input()) # e.g. L=5
for i in range(0, L): # outer: rows 行
for j in range(0, L): # inner: cols 列
print("*", end="")
print() # newline 換行
print("*", end="") — suppresses newline, prints on same line (不換行).
end="" 參數取消換行，使 * 在同一行輸出。
Summary 總結 (Page 2)
Nested loop = loop inside loop (嵌套循環 = 循環中的循環).
Each outer iteration → inner loop runs COMPLETELY (每次外循環 → 內循環完整執行一遍).
Total reps = outer × inner (總次數 = 外 × 內). Clock analogy: minute=outer, second=inner.
print("*", end="") keeps output on same line (end="" 使輸出在同一行).
range(0, 2)
= 0, 1 → 2 reps
range(0, 3)
= 0,1,2 → 3 reps
記住！
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Key Term 關鍵詞
2D Array 二維陣列
rows × columns 的格狀結構
Q: How to access num[i][j]?
如何存取 num[i][j]？
Key Term 關鍵詞
Initialisation 初始化
在存取前先建立列表並賦予預設值
Q: Why must we initialise?
為何必須初始化？
Key Term 關鍵詞
Index 索引
num[i][j]
i = row 行, j = column 列
Tip 提示
Outer loop → rows 行
Inner loop → cols 列
外循環 → 行
內循環 → 列
See Also 參見
→ Page 2: Nested loops
→ 第2頁：嵌套循環
→ Page 4: Applications
→ 第4頁：應用
3.3 Accessing 2D Array with Nested for Loop 用嵌套循環存取二維陣列
Use nested for loop to access items in a 2D array one by one.
用嵌套 for 循環逐一存取二維陣列中的項目。
# Access all items row by row 逐行存取
for i in range(0, 3): # rows i=0,1,2
for j in range(0, 4): # cols j=0,1,2,3
num[i][j] = i * j # assign value 賦值
Note: Swapping outer/inner loop order changes the traversal direction but not the result when accessing all items.
注意：交換內外循環的順序會改變遍歷方向，但存取所有項目的結果不變。
Initialisation of 2D List 二維列表的初始化
Before accessing items, must create the list and assign default values = initialisation 初始化.
在存取項目前，必須先建立列表並賦予預設值，此過程稱為初始化。
# Initialise 3×6 list with 0s 初始化 3×6 列表
num = []
for i in range(0, 3): # ① add rows 增加行
num = num + [[]]
for j in range(0, 6): # ② add cols 增加列
num[i] = num[i] + [0]
num[i][j] = i * j # assign 賦值
Example 3.2 — Bookshelf Total 書架總書數
B[i][j] stores number of books in storage unit (i,j). Find total books.
B[i][j] 儲存位置 (i,j) 的書本數量，求總書數。
total = 0
for i in range(0, 3): # 3 rows 行
for j in range(0, 6): # 6 cols 列
total = total + B[i][j]
print(total)
Pattern: outer loop = rows (行), inner loop = columns (列). Works for any m×n array.
模式：外循環 = 行，內循環 = 列。適用於任何 m×n 陣列。
2D Array Index Diagram 二維陣列索引圖
| i \ j | j=0 | j=1 | j=2 | j=3 |
| i=0 | num[0][0] | num[0][1] | num[0][2] | num[0][3] |
| i=1 | num[1][0] | num[1][1] | num[1][2] | num[1][3] |
| i=2 | num[2][0] | num[2][1] | num[2][2] | num[2][3] |
Summary 總結 (Page 3)
2D array accessed by num[i][j] — i=row (行), j=col (列). Nested for: outer=rows, inner=cols.
Initialisation (初始化) = create list + assign default values BEFORE accessing items.
Pattern for sum: total=0 → nested for → total = total + array[i][j] → print(total).
num[i][j]
i = row 行
j = col 列
Always init first!
先初始化！
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Key Concept 核心概念
Integrated Control
綜合控制結構
三種結構可組合使用
Q: Name all 3 structures.
說出三種控制結構。
Key Term 關鍵詞
Control Flow
控制流程
語句執行的順序
Key Term 關鍵詞
Multi-layered Nested
多層嵌套結構
控制結構 + 數據結構組合
Q: How many while loops
in Example 3.3?
例3.3有幾個while循環？
Warning ⚠
Array index out
of range! 索引越界！
Always check:
0 ≤ i ≤ max
0 ≤ j ≤ max
See Also 參見
→ Page 3: 2D arrays
→ Page 5: Comparison
→ 第3頁：二維陣列
→ 第5頁：演算法比較
3.4 Integrated Application 綜合應用
Control structures can be combined with data structures (1D & 2D arrays) to form multi-layered nested structures.
控制結構可與數據結構（一維及二維陣列）組合，形成多層嵌套結構。
| Structure 結構 | Role 作用 |
|---|
| Sequence 順序 | Execute statements in order 按順序執行 |
| Selection 選擇 | Decide which path 決定執行哪個分支 |
| Iteration 迭代 | Repeat statements 重複執行語句 |
Example 3.3 — Agricultural Robot 農業機器人 (while + for + if)
F[y,x] = 1 (sow 播種) or 0 (skip 跳過) in 10×10 farm grid.
F[y,x] 儲存 10×10 農場格的播種指令（1=播種，0=跳過）。
y ← 0
while y < 10 do # outer while 外while
x ← 0
while x < 10 do # inner while 內while
for i from y to y+n-1 # nested for 嵌套for
for j from x to x+n-1
if i <= 9 and j <= 9 then # boundary check 邊界檢查
F[i,j] ← 1
x ← x + (n*2)
y ← y + (n*2)
Analysis 分析:
- Inner nested for (lines 5-7): creates n×n square at position (x,y)
嵌套 for 循環：在位置 (x,y) 建立 n×n 的播種方塊
- Inner while (lines 4-9): creates a row of n×n squares, spaced n apart
內 while 循環：建立一行 n×n 方塊，間距為 n
- Outer while (lines 2-10): creates multiple rows of squares
外 while 循環：建立多行方塊
- Boundary check: if i ≤ 9 and j ≤ 9 prevents index out of range
邊界檢查：防止索引越界（陣列索引不可超出範圍）
Summary 總結 (Page 4)
Any complex program = combination of Sequence + Selection + Iteration (順序+選擇+迭代).
Pair with 1D/2D arrays for multi-layered nested structures (多層嵌套結構).
ALWAYS check array index boundaries (始終檢查陣列索引邊界): 0 ≤ index ≤ max.
while+for+if nested together → powerful solution design (強大的解決方案設計).
Boundary check!
邊界檢查！
if i<=9 and
j<=9 then...
防止索引越界！
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Key Term 關鍵詞
Algorithm Comparison
演算法比較
選擇最正確、適切的解決方案
Q: 5 criteria for comparison?
比較演算法的5個標準？
Key Term 關鍵詞
Time Complexity
時間複雜度
步驟越多 → 時間越長
Q: More steps = ?
步驟更多 = ？
Key Term 關鍵詞
Space Complexity
空間複雜度
佔用資源越多 → 空間複雜度越高
5 Criteria 五個標準
① Correctness 正確性
② Appropriateness 適切性
③ Steps/Time 步驟/時間
④ Resource usage 資源使用
⑤ Complexity 複雜度
⑥ Data structures 數據結構
Trade-off 取捨
Q: Can an algorithm be fast
but use more memory?
演算法可以快但
佔更多記憶體嗎？
3.5 Algorithm Comparison 演算法比較
A problem can often be solved by several algorithms — each with merits & demerits.
一個問題通常可用多個演算法解決，各有優缺點。
Compare algorithms to select the most correct and appropriate solution.
比較演算法以選擇最正確、最適切的解決方案。
Comparison Criteria 比較標準
| Criterion 標準 | Meaning 含義 | Impact 影響 |
|---|
Correctness
正確性 |
Produces correct output
輸出正確結果 |
Most important
最重要 |
Appropriateness
適切性 |
Handles all cases
處理所有情況 |
Covers edge cases
涵蓋邊界情況 |
Steps / Time
步驟/時間 |
Number of operations
運算步驟數量 |
↑ steps → ↑ time
步驟多→時間長 |
Resource usage
資源使用 |
Memory, CPU used
記憶體、CPU用量 |
Space complexity
空間複雜度 |
Data structures
數據結構 |
Type of arrays used
使用的陣列類型 |
Affects complexity
影響複雜度 |
Checkpoint 3.4 — Missing Item (3 Algorithms) 缺失項目（三種演算法）
Box A contains items A to A+9. Array C has 9 items (one missing). Find missing item.
箱子 A 應有 A 至 A+9 共10件物品。陣列 C 有9件，找出缺失的一件。
| Algorithm 演算法 | Method 方法 | Advantage 優點 |
|---|
| Algorithm 1 |
Sum expected − sum actual
預期總和 − 實際總和 |
Fewer steps 步驟少
More efficient 更高效 |
| Algorithm 2 |
For each expected item,
search in array
逐一搜尋 |
Can find duplicates
可找重複項 |
| Algorithm 3 |
Mark found items in
boolean array
用布林陣列標記 |
More readable
可讀性更高 |
Algorithm 1 most efficient (fewer steps). Algorithms 2 & 3 can handle special case: finding duplicates (重複項).
演算法1最高效（步驟最少）。演算法2和3可處理特殊情況：找出重複項目。
Summary 總結 (Page 5)
Compare algorithms by: Correctness, Appropriateness, Steps/Time, Resource usage, Complexity, Data structures (正確性、適切性、步驟/時間、資源、複雜度、數據結構).
Time complexity (時間複雜度): more steps → longer time. Space complexity (空間複雜度): more resources → higher.
Trade-off: fast algorithm may use more memory (快速演算法可能佔用更多記憶體).
Alg 1: sum trick
最少步驟！
Alg 2 & 3: can
find duplicates
可找重複項！
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Review 複習
Chapter Glossary
章節詞彙
Q: Define all 4 terms.
定義全部4個詞彙。
Key Pattern 重要模式
Sum of 2D Array
二維陣列求和
total=0 → nested for → accumulate
Key Pattern 重要模式
Draw * Pattern
繪製 * 圖案
outer=rows, inner=cols
print("*",end="")
Q: How to print a triangle?
如何輸出三角形？
Key Pattern 重要模式
Search in 2D
二維搜尋
nested for + if condition
嵌套for + 條件判斷
Exam Tip 考試技巧
Always trace with
trace table! 用追蹤表！
Check i,j,output
each iteration
每次迭代檢查i,j,輸出
Chapter 3 Glossary 詞彙表
| English Term | Chinese 中文 | Definition 定義 |
|---|
| initialisation | 初始化 | Create list & assign default values before access 存取前建立列表並賦予預設值 |
| nested conditional statement | 嵌套條件語句 | if inside another if/else 在另一個if/else中的if |
| nested loop | 嵌套循環 | Loop inside another loop 循環中的循環 |
| nested structure | 嵌套結構 | Structure placed inside another 結構放入另一結構 |
Key Code Patterns 重要代碼模式
① Sum 2D Array 二維陣列求和
total = 0
for i in range(0,R):
for j in range(0,C):
total+=arr[i][j]
print(total)
② Draw Rectangle 繪製矩形
for i in range(0,H):
for j in range(0,W):
print("*",end="")
print()
③ Draw Triangle 繪製三角形
for i in range(0,H):
for j in range(0,i+1):
print("*",end="")
print()
④ Search 2D 二維搜尋
found = False
for i in range(0,R):
for j in range(0,C):
if arr[i][j]==x:
found=True
Chapter 3 Mind Map 思維導圖
Ch.3 Advanced
Control Structures
高級控制結構
3.1 Nested if
嵌套條件語句
3.2 Nested Loops
嵌套循環
3.3 2D Arrays
二維陣列
3.4 Integration
綜合應用
3.5 Comparison
演算法比較
Summary 總結 (Page 6) — Full Chapter Review 全章回顧
Ch.3: Nested if (嵌套條件) → Nested loops (嵌套循環) → 2D arrays (二維陣列) → Integration (綜合) → Comparison (比較).
Key: total=outer×inner reps; init 2D list before use; boundary check always; compare algorithms by 5 criteria.
Exam: use trace table (追蹤表) to dry-run nested loops step by step!
EC3 done! 完成！
Review all 6 pages
before exam!
考試前溫習
全部6頁！
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