2D arrays: a table with two indexes

Click cells, trace nested loops line by line, and see the exam questions OCR asks. Switch between GCSE (J277, topic 2.2.3) and A Level (H446, topic 1.4.2), and between OCR's exam notation and Python.

Quiz me

A 1D array is a list with one index. A 2D array is a table: one identifier, two indexes, usually written name[row, column].
Indexes start at 0. An array declared [4, 5] has rows 0 to 3 and columns 0 to 4, and holds 4 × 5 = 20 elements.
Every element has the same data type, and the size is fixed when it is declared.
To visit every element you need nested loops: the outer loop chooses the row, the inner loop walks along it.
OCR allows arrays of up to three dimensions: array cube[2, 3, 4] has 2 × 3 × 4 = 24 elements. The same ideas apply with one more index and one more loop.

The array

Change the selected element

Which index comes first?

Reading an index

The first number picks the row (count down from 0).
The second number picks the column (count across from 0).
Where they cross is the element. name[2, 3] is the fourth element of the third row, because counting starts at 0.
If a question defines its own order, such as "grid[x, y] where x is the column", follow the question.

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Program

Highlighted line is the one being executed

How nested loops visit a 2D array

The outer loop sets the row index once per row.
For each row, the inner loop runs all the way through the columns.
So the cells are visited row by row: [0,0], [0,1], … [0,C−1], then [1,0], [1,1], …
Swap the loops (columns outside, rows inside) and the same cells are visited column by column instead.
Total iterations of the inner body = rows × columns.

Live grid

faded = already visited, purple = this step

How a 2D array is stored

Memory is one-dimensional: a numbered sequence of bytes. A 2D array is stored as one contiguous block, row after row (row-major order). The compiler turns the two indexes into one address with a formula, which is why every element can be reached in the same, constant time.

Base address (address of element [0,0])

Size of each element

Points the mark scheme looks for

Arrays are static: the size is fixed at declaration and the memory is allocated in one block, so it cannot grow at run time.
Elements are stored contiguously (next to each other), all of the same type, so each occupies the same number of bytes.
Address of [r, c] = base + (r × columns + c) × element size. Any element is found with one calculation: direct access, O(1).
Three dimensions: [i, j, k] in an array [A, B, C] is element number (i × B + j) × C + k.
When an array is passed to a subroutine it is normally passed by reference: the subroutine receives the address, not a copy, so changes affect the original and nothing is copied. Pass by value would copy the whole block.

Click an element to see where it lives

The same elements as one block of memory

Quick quiz

Questions use the array currently shown on the Explore tab. Give an index as two numbers, for example 2,3.

Score: 0 / 0

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Exam tips and common mistakes

Indexes start at 0. array seats[4, 5] has rows 0 to 3 and columns 0 to 4. seats[4, 5] does not exist: that is an out-of-bounds error, the single most common slip.
Say which index is which. Unless the question tells you otherwise, treat the first index as the row and the second as the column, and keep to it throughout your answer. If the question defines grid[x, y] with x across, follow the question.
Loop bounds. In OCR notation for c = 0 to 4 runs five times and includes 4. In Python range(5) runs five times and excludes 5. Writing to 5 for five columns goes out of bounds.
Close the inner loop first. next c comes before next r. In Python this is indentation; in OCR notation it is the order of the next lines.
Accumulators. Set total = 0 before the loops. A row total must be reset to 0 inside the outer loop, once per row, not once at the start.
Output inside or outside the loop? To print every element, the print goes inside the inner loop. To print one total, it goes after both loops. Marks are lost for printing a running total 20 times.
Trace tables. Use a column for r, a column for c, a column for the element being read and one for each variable that changes. Only write a value when it changes.
Number of elements = rows × columns. A 3D array multiplies all three sizes.
Python's lists of lists (grid[r][c]) are accepted in the exam as a 2D array. Do not mix the notations in one answer: grid[r, c] in OCR style, grid[r][c] in Python.
A Level: be ready to explain that arrays are static, contiguous and the same type throughout, to give the address formula, and to say why an array parameter is passed by reference.

The array

Reading an index

Click any element

Program

How nested loops visit a 2D array

Live grid

How a 2D array is stored

Points the mark scheme looks for

Click an element to see where it lives

Exam-style questions

Quick quiz