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Algorithms Term 3

Trace Tables

Problem Statement

Complete a trace table for the following algorithm that calculates the factorial of a number. Use the input value n = 4. 

BEGIN Factorial
    INPUT n
    result = 1
    counter = 1

    WHILE counter <= n
        result = result * counter
        counter = counter + 1
    ENDWHILE

    OUTPUT result
END

Step-by-Step Solution

1

Identify all variables

List every variable used in the algorithm. These become your column headers.

Variables found:

  • n - the input number
  • result - stores the factorial being calculated
  • counter - loop counter

Also track:

  • counter <= n - the loop condition (TRUE/FALSE)
  • Output - what gets displayed
2

Set up the trace table

Create columns for each variable, the condition, and output. Add a row number column.

Step n result counter counter <= n Output
...
3

Trace through initialization

Record the initial values before the loop starts. 

INPUT n          // n = 4
result = 1       // result = 1
counter = 1      // counter = 1
Step n result counter counter <= n Output
1 4 1 1 - -
4

Trace each loop iteration

For each iteration: check condition, execute body, update variables.

Iteration 1:

  • Check: Is 1 <= 4? TRUE (enter loop)
  • result = 1 * 1 = 1
  • counter = 1 + 1 = 2

Iteration 2:

  • Check: Is 2 <= 4? TRUE (continue loop)
  • result = 1 * 2 = 2
  • counter = 2 + 1 = 3

Iteration 3:

  • Check: Is 3 <= 4? TRUE (continue loop)
  • result = 2 * 3 = 6
  • counter = 3 + 1 = 4

Iteration 4:

  • Check: Is 4 <= 4? TRUE (continue loop)
  • result = 6 * 4 = 24
  • counter = 4 + 1 = 5

Exit check:

  • Check: Is 5 <= 4? FALSE (exit loop)
  • OUTPUT result → 24

Complete Trace Table

Step n result counter counter <= n Output
1 (init) 4 1 1 - -
2 4 1 1 TRUE -
3 4 1 2 - -
4 4 1 2 TRUE -
5 4 2 3 - -
6 4 2 3 TRUE -
7 4 6 4 - -
8 4 6 4 TRUE -
9 4 24 5 - -
10 4 24 5 FALSE -
11 4 24 5 - 24

Result: The factorial of 4 is 24 (4! = 4 × 3 × 2 × 1 = 24)

Tips for Trace Tables

  • ✓ Only update values that actually change in each step
  • ✓ Check the loop condition BEFORE each iteration
  • ✓ Use a dash (-) for values that haven't been set yet or don't apply
  • ✓ Verify your final answer makes sense (4! should equal 24)

Common Mistakes to Avoid

  • ✗ Forgetting to check the condition before each loop iteration
  • ✗ Using the OLD value of a variable after it's been updated
  • ✗ Missing the final condition check that exits the loop
  • ✗ Not including all variables (especially the loop counter)