A-Level Computer Science / Unit 10: Organising Data in Programs

10.2.2 Two-Dimensional Arrays and Nested Loops

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10.2.2 Two-Dimensional Arrays and Nested Loops

A two-dimensional array stores values in a rectangular arrangement. Each element is selected by an index pair: one index identifies the row and the other identifies the column. Nested loops allow a program to visit every position systematically.

By the end of this section, you should be able to:

  • Decide when a problem is better represented by a 2D array than a 1D array.
  • Interpret row and column bounds in a 2D array declaration.
  • Calculate the number of rows, columns and total elements.
  • Access, assign, input and output an individual cell using an index pair.
  • Use nested loops to initialise, read and display a complete 2D array.
  • Trace the order in which nested loops move through a grid.

When Is a 2D Array Suitable?

Imagine a greenhouse divided into planting rows and sensor columns. One soil-moisture reading is stored for every intersection. A single position cannot be identified by only one number: the program needs both the planting row and the sensor column.

Data to store Likely structure Reason
One reading for each of eight checkpoints 1D array Each value needs one checkpoint index.
One reading for each row-column position in a greenhouse 2D array Each value needs a row index and a column index.
One passenger name for each seat in several rows 2D array The seat is located by row and seat number.
Two-dimensional array: an array whose elements are selected using two indexes, commonly written as row first and column second.
Exam tip: Justify a 2D array by referring to the two independent positions needed to locate each value, not simply by saying that there are “many values”.

Rows, Columns and Index Pairs

The following grid uses row indexes 2 to 5 and column indexes 10 to 14. These index values describe positions; they do not have to begin at 0 or 1.

Row \ Column 10 11 12 13 14
24852455750
36155596354
44649515853
56460566247

The value 59 is located at row 3, column 12, so it is addressed as SoilMoisture[3, 12].

Index pair: the two index values used to identify one element in a 2D array.
Common mistake: SoilMoisture[3, 12] does not mean “element 3 then element 12”. It identifies one cell at the intersection of row 3 and column 12.

Declaring a 2D Array

A declaration includes one bound range for each dimension and one data type for all stored elements.

General form:
DECLARE <identifier> : ARRAY[<rowLower>:<rowUpper>, <columnLower>:<columnUpper>] OF <dataType> 
DECLARE SoilMoisture : ARRAY[2:5, 10:14] OF INTEGER
Declaration part Meaning
SoilMoistureThe identifier for the complete 2D array.
2:5Valid row indexes are 2, 3, 4 and 5.
10:14Valid column indexes are 10, 11, 12, 13 and 14.
INTEGEREvery cell stores a whole-number moisture reading.

Calculate the size

Number of rows    = 5 - 2 + 1 = 4
Number of columns = 14 - 10 + 1 = 5
Total elements    = 4 * 5 = 20
For a 2D array, calculate the size of each dimension separately, then multiply the two results.

Accessing and Updating One Cell

In this course, the first index identifies the row and the second identifies the column.

Assign a value

SoilMoisture[3, 12] ← 59

Save input directly into a cell

INPUT SoilMoisture[5, 14]

Output one cell

OUTPUT SoilMoisture[2, 11]

Update an existing value

SoilMoisture[4, 13] ← SoilMoisture[4, 13] + 2
ExpressionMeaning
SoilMoistureThe complete 2D array.
SoilMoisture[3, 12]One element at row 3, column 12.
SoilMoisture[Row, Column]The cell selected by the current values of two index variables.
A reference such as SoilMoisture[3] is incomplete because the column index is missing.

Why Nested Loops Are Needed

One loop can visit every row or every column, but not both dimensions by itself. A nested loop places one loop inside another so that every row-column combination is reached.

Nested loop: a loop inside another loop. The inner loop completes all of its iterations for each single iteration of the outer loop. 
FOR Row ← 2 TO 5
    FOR Column ← 10 TO 14
        OUTPUT SoilMoisture[Row, Column]
    NEXT Column
NEXT Row

For row 2, the inner loop visits columns 10 to 14. It then starts again at column 10 when the outer loop advances to row 3.

Match the loop limits to the declared bounds. Do not replace the actual index range with 1 TO numberOfElements unless the declaration uses those indexes.

Processing a Complete Grid

Initialise every element

FOR Row ← 2 TO 5
    FOR Column ← 10 TO 14
        SoilMoisture[Row, Column] ← 0
    NEXT Column
NEXT Row

Input every reading

FOR Row ← 2 TO 5
    FOR Column ← 10 TO 14
        INPUT SoilMoisture[Row, Column]
    NEXT Column
NEXT Row

Display the array as a grid

FOR Row ← 2 TO 5
    FOR Column ← 10 TO 14
        OUTPUT SoilMoisture[Row, Column]
    NEXT Column
    OUTPUT Newline
NEXT Row

The inner loop outputs all values from one row. OUTPUT Newline is placed after the inner loop so that the next array row begins on a new output line.

Calculate the total of all cells

TotalMoisture ← 0

FOR Row ← 2 TO 5
    FOR Column ← 10 TO 14
        TotalMoisture ← TotalMoisture + SoilMoisture[Row, Column]
    NEXT Column
NEXT Row

OUTPUT TotalMoisture

Tracing the Traversal Order

With rows in the outer loop and columns in the inner loop, the program completes one row before moving to the next. The first seven positions are:

StepRowColumnElement visited
1210SoilMoisture[2, 10]
2211SoilMoisture[2, 11]
3212SoilMoisture[2, 12]
4213SoilMoisture[2, 13]
5214SoilMoisture[2, 14]
6310SoilMoisture[3, 10]
7311SoilMoisture[3, 11]
Common mistake: The column variable does not continue from 14 to 15. It returns to its lower bound when the outer loop advances to the next row.

Reversing the loops would visit the same cells in a different order: one complete column at a time. The required order depends on how the data should be processed or displayed.

Interactive: 2D Array Grid Visualiser

Use the existing widget to select a row-column position, change one cell and trace the order used by nested loops. The small demonstration grid keeps the traversal visible while using the same principles as a larger 2D array.

Select a cell

Pseudocode focus 
Grid[1, 1] ← blank

Choose a row and column to see how a single cell is addressed.

Columns
Rows

Nested loop trace

Press Start trace to see each row-column position visited in order.

Common Mistakes and Misconceptions

  • Using only one index for an element in a 2D array.
  • Reversing the intended row-column order.
  • Assuming that row and column indexes must begin at 0 or 1.
  • Calculating the total size without first finding the size of each dimension.
  • Using nested-loop limits that do not match the declaration.
  • Placing OUTPUT Newline inside the inner loop, which puts every cell on a separate line.
  • Forgetting that the inner loop restarts for every outer-loop iteration.
  • Confusing a 2D array with an array of records. Arrays of records were covered in Section 10.1.2.

Practice

Task 1: Interpret a declaration

DECLARE SeatState : ARRAY[4:9, 20:27] OF CHAR

  1. State the lower and upper row bounds.
  2. State the lower and upper column bounds.
  3. Calculate the number of rows.
  4. Calculate the number of columns.
  5. Calculate the total number of elements.
  6. Write the expression for row 6, column 24.

Task 2: Write pseudocode

  1. Declare Rainfall with rows 1 to 6 and columns 3 to 9, storing REAL values.
  2. Assign 18.6 to row 4, column 7.
  3. Write nested loops that set every cell to 0.0.
  4. Write nested loops that input every value.
  5. Write nested loops that output each array row on a separate line.

Task 3: Trace nested loops

FOR Row ← 1 TO 2
    FOR Column ← 5 TO 7
        OUTPUT Grid[Row, Column]
    NEXT Column
NEXT Row
  1. List the six elements in the exact order visited.
  2. How many times does the inner loop execute in total?
  3. What value does Column restart with when Row becomes 2?

Review

QuestionStrong answer should include
What is a 2D array?A rectangular arrangement of elements selected using two indexes.
How is one element addressed?Use the array identifier and an index pair, normally row then column.
How is the total size calculated?Calculate each dimension with upper − lower + 1, then multiply.
Why are nested loops useful?They generate every combination of valid row and column indexes.
Where should a new line be output when displaying a grid?After the inner column loop, once a complete row has been output.
What happens to the inner loop for each new row?It restarts from its lower bound and completes its full range again.
Final exam tip: Include complete index notation and make every loop bound agree with the declaration. Correct array ideas can still lose marks when one index, one bound or one loop limit is missing.