# Lecture 6: Use ranges and invariants to control repetition ## for loops, nested loops, and invariants **Instructor:** [Dr. Debasish Pattanayak](https://drdebmath.github.io)

## The counter has one clear valid range. · Print a multiplication table without guessing bounds A counted loop names its start, continuation rule, and update in one place; the loop invariant explains what has already been printed. ~~~cpp #include <iostream> int main() { int number; std::cin >> number; for (int multiplier = 1; multiplier <= 10; ++multiplier) { std::cout << number << " × " << multiplier << " = " << number * multiplier << '\n'; } } ~~~ **Takeaways** - The counter has one clear valid range. - The update moves toward termination. - The invariant describes completed rows.

## A for loop makes initialization, bound, and update explicit ~~~cpp for (int step = 1; step <= 5; ++step) { std::cout << step << ' '; } ~~~ Read it as initialize → test → body → update → test again.

## The general form of a for and a nested for ~~~cpp for (<initialization>; <condition>; <update>) { // body, repeated while <condition> stays true } for (<outer initialization>; <outer condition>; <outer update>) { for (<inner initialization>; <inner condition>; <inner update>) { // runs once for every pair of outer and inner values } } ~~~ | Part | Meaning | | --- | --- | | `<initialization>` | Runs once, before the first test. A counter declared here is visible only inside the loop. | | `<condition>` | Tested before every iteration, including the first. | | `<update>` | Runs after every iteration, before the next test. | | nesting | The inner loop completes in full for each single iteration of the outer loop. | The inner loop may depend on the outer counter; that is what makes triangles and tables.

## A correct counter states its valid range For five zero-based positions, valid indices are 0, 1, 2, 3, 4. ~~~cpp for (int i = 0; i < 5; ++i) { std::cout << i << ' '; } ~~~ The condition directly states the upper boundary is excluded.

## Three clauses arrive at rehearsal together <figure class="course-meme-figure"> <img src="assets/lecture-memes/l06/for-parts.png" alt="A conductor cues a tiny marching loop with a start signal, a boundary gate, and a step drumbeat all in one coordinated scene — visual point: Start, check, step — the rhythm stays where the reader can see it." width="900" height="506" loading="lazy"> <figcaption> <p class="course-meme-setup"><strong>Setup</strong> The loop’s three clauses arrive at rehearsal.</p> <p class="course-meme-punchline"><strong>Punchline</strong> Start, check, step — the rhythm stays where the reader can see it.</p> </figcaption> </figure>

## Nested loops enumerate every coordinate or combination ~~~cpp for (int row = 0; row < 3; ++row) { for (int col = 0; col < 4; ++col) { std::cout << row << ',' << col << '\n'; } } ~~~ The inner loop completes every column for one row.

## Game evolution: Rendering turns state into a board | Today's concept | Exact game part enabled | |---|---| | nested for loops | visit every 2D position and choose the symbol to print | ~~~cpp #include <iostream> int main() { int actorRow=1, actorColumn=3; for (int row=0; row<3; ++row) { for (int column=0; column<7; ++column) { if (row==actorRow && column==actorColumn) std::cout << 'S'; else std::cout << '.'; } std::cout << '\n'; } } ~~~ **Loop invariant:** before row <code>r</code>, every cell in earlier rows has been rendered exactly once. **Next unlock · L07:** name scoring and movement calculations as testable functions.

## A loop invariant states what stays true after each iteration Example invariant: > Before iteration <code>i</code>, positions 0 through <code>i - 1</code> have already been processed. Check the statement before the first iteration, after one update, and at termination.

## The sixth tile is feeling optimistic <figure class="course-meme-figure"> <img src="assets/lecture-memes/l06/exact-range.png" alt="A student marks five floor tiles from the first tile through the last valid tile while a sixth tile is visibly outside the safe boundary — visual point: Write the valid range first; let the bound exclude what does not exist." width="900" height="506" loading="lazy"> <figcaption> <p class="course-meme-setup"><strong>Setup</strong> Five positions exist. The sixth tile disagrees.</p> <p class="course-meme-punchline"><strong>Punchline</strong> Write the valid range first; let the bound exclude what does not exist.</p> </figcaption> </figure>

## Choose for, while, or do-while from the stopping rule - <code>while</code>: stop when an event or changing condition says so. - <code>do-while</code>: perform once, then decide. - <code>for</code>: traverse a known counter range. The choice communicates intent; any loop can still be written incorrectly.

## Boundary tests expose missing or duplicated iterations ~~~cpp for (int i = 0; i <= 5; ++i) // six values for (int i = 5; i >= 0; ++i) // moves away from stopping ~~~ These are diagnosis examples. Do not run the second loop unchanged.

## The inner loop promises it will be quick <figure class="course-meme-figure"> <img src="assets/lecture-memes/l06/nested-coordinates.png" alt="A tiny robot sweeps every cell of a rectangular grid, completing one row before moving to the next — visual point: Four columns disagree, and the outer loop is still waiting." width="900" height="506" loading="lazy"> <figcaption> <p class="course-meme-setup"><strong>Setup</strong> The inner loop says it will be quick.</p> <p class="course-meme-punchline"><strong>Punchline</strong> Four columns disagree, and the outer loop is still waiting.</p> </figcaption> </figure>

## A correct loop states its range, invariant, and progress - Put initialization, test, and update together for counted work. - Nested loops traverse combinations such as rows × columns. - Write valid values before writing a bound. - Use an invariant to explain what is already correct.

## Verify Print a multiplication table without guessing bounds with a claim, trace, boundary, and improvement Use the practical example as a small experiment. Before you leave, make the reasoning visible: - **Claim:** The counter has one clear valid range. - **Trace:** name the state that changes and the observation that would confirm it. - **Boundary:** choose one smallest, largest, empty, or invalid input that could expose a defect. - **Improvement:** explain one change that would make the solution clearer or safer. ### Exit ticket Choose one problem to solve now and two to attempt in the lab: - Print the first n odd numbers and their sum. - Draw an r-by-c rectangle using nested loops. - Trace a loop with <= changed to < and explain the difference.

## Explain Print a multiplication table without guessing bounds without opening the code Explain today’s idea to a partner without opening the code: 1. State the input, output, and one rule the solution must preserve. 2. Point to the operation that makes progress or changes the state. 3. Name the first test you would run after changing the implementation. If the explanation needs “and then another unrelated thing,” split the design before you code.

## Solve five problems to transfer Print a multiplication table without guessing bounds **IC151 lab practice · 5 problems** 1. Print the first n odd numbers and their sum. 1. Draw an r-by-c rectangle using nested loops. 1. Trace a loop with <= changed to < and explain the difference. 1. Rewrite a counted while loop as a for loop. 1. Find and repair three non-terminating or off-by-one loops. Reaching this set marks the lecture as studied in this browser. [Open the IC151 lab setup and batch schedule](ic151.html).

## Did you know? - Tony Hoare introduced the logic of loop invariants in 1969. He also invented quicksort, and later called the null reference his "billion-dollar mistake". - Edsger Dijkstra's 1968 letter "Go To Statement Considered Harmful" is why we write structured loops instead of jumps. - Dijkstra also argued, in a 1982 note, that numbering should start at zero, which is why `for (int i = 0; i < n; ++i)` looks the way it does.

## Visit each cell of a small room and print an ASCII frame. **Studio thread:** Sweep and render a grid | Ask | Today’s answer | |---|---| | **Problem** | Visit each cell of a small room and print an ASCII frame. | | **Model** | A room is rows × columns; the current pair identifies one cell. | | **Represent** | Nested counters represent coordinates; characters represent visible cell states. | | **Solve** | Enumerate every coordinate with nested for loops. | | **Verify** | State the processed-prefix invariant and count expected visits before running. | | **Improve** | Separate the sweep order from rendering and diagnose off-by-one bounds. | **Technique lens:** systematic enumeration **Compare:** row-major versus column-major traversal