Algorithms, efficiency, logic errors and trace tables

What an algorithm is

An algorithm is a finite sequence of unambiguous steps that solves a problem or completes a task. Good algorithms are:

• finite: they stop;
• unambiguous: each step has one meaning;
• clear about inputs and outputs;
• correct for all valid inputs, including edge cases.

Representations: written description, flowchart (terminator for start and end, rectangle for a process, parallelogram for input/output, diamond for a decision, arrows for flow), and code (Python in the Pearson papers). Algorithms use three constructs: sequence, selection (if) and iteration (loops).

Efficiency

Efficiency is about how the number of steps (time) or amount of memory grows as the input gets bigger. For a list of n items:

• linear search may need up to n comparisons;
• binary search needs at most about log2 n comparisons (for example 10 for 1000 items);
• bubble sort and insertion sort may need about n²/2 comparisons.

A more efficient algorithm matters most when n is large.

Logic errors and trace tables

A logic error makes a program run but give the wrong result, for example using < instead of <=, starting a counter at the wrong value, or updating a variable in the wrong place. (A syntax error breaks the rules of the language and stops it running.)

A trace table records the value of each variable (and any output) every time it changes, line by line. Use one row per change, and only fill a cell when that variable changes. Tracing reveals logic errors by showing where values go wrong.

Example trace

total = 0
i = 1
while i <= 7:
    total = total + i * i
    i = i + 2
print(total)

The loop stops when i reaches 9. Output: 84

Worked example

This code should output the largest value in nums but outputs the wrong answer for [-4, -9, -2]. Find and fix the logic error. ```python biggest = 0 for n in nums: if n > biggest: biggest = n print(biggest) ```

1. Every value is negative, so n > 0 is never true.
2. biggest stays 0, which is not in the list.
3. Fix: start with the first item, biggest = nums[0].

Answer: Initialise biggest = nums[0]; output is then -2.

Check yourself

1. Trace: total = 0, i = 1; while i <= 7: total = total + i * i; i = i + 2. What is printed after the loop?
   Show answer

   84. i takes 1, 3, 5, 7: 1 + 9 + 25 + 49 = 84.

2. In the loop i = 1; while i <= 7: (body); i = i + 2, how many times does the loop body run?
   Show answer

   4. i = 1, 3, 5, 7: 4 times.

3. Trace: x = 2; for k in range(3): x = x * 3 - 1. What is x afterwards?
   Show answer

   41. 2 becomes 5, then 14, then 41.

4. Which flowchart symbol represents a decision?
   • Rectangle
   • Parallelogram
   • Rounded rectangle (terminator)
   • Diamond
   Show answer

   Diamond. A decision has a yes and a no exit.

5. A program to count the numbers 1 to 10 uses while n < 10, starting at n = 1, and counts only 9 numbers. What type of error is this?
   • Syntax error
   • Logic error
   • Runtime error caused by dividing by zero
   • Hardware error
   Show answer

   Logic error. It runs but gives the wrong result; the condition should be n <= 10.

6. Approximately how many comparisons does binary search need, at most, to search a sorted list of 1024 items?
   Show answer

   11. Each comparison halves the list: 1024 halves 10 times to 1 item, so at most 11 comparisons (log2 1024 = 10, plus the final check).

7. Which property must every algorithm have?
   • It must use a loop
   • It must stop after a finite number of steps
   • It must be written in Python
   • It must sort a list
   Show answer

   It must stop after a finite number of steps. An algorithm that never ends does not solve the problem.

8. Trace: c = 0; for v in [4, 7, 10, 13, 16]: if v % 2 == 1: c = c + v. What is c?
   Show answer

   20. Only 7 and 13 are odd: 7 + 13 = 20.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Computer Science specification (2026)
• Pearson Edexcel IAL Computer Science IAS sample assessment materials (Unit 1, WCP01)

Last updated 8 November 2026. Spotted a mistake? Tell us.