Loop algorithms and trace tables

A teacher has to add up the scores of 30 students. Instead of writing “add the next score to the sum” 30 times, it is enough to write it once and say “repeat 30 times”. An algorithm with repeated steps is a loop algorithm. Loops were the most frequent topic of the 2025–2026 entrance exams: Python tasks labelled “Loop statement”, plus flowchart tasks on the number of iterations, two consecutive loops and reverse questions such as “which input must be given so that…”. In this lesson you will learn to read loops in flowcharts and to run them without mistakes using a trace table.

Loops and their kinds

• Counter loop — the number of repetitions is known in advance: the loop variable runs from a start value to an end value with a fixed step. In Python: for i in range(1, 11).
• Loop with the condition first — the condition is checked before the body, and the body is repeated while the condition is true. If the condition is false from the start, the body runs zero times. In Python: while.
• Loop with the condition last — the body runs first and then the condition is checked, so the body runs at least once. Python has no separate statement for it; it is written with while True: and break.

for i from 1 to 5
    S = S + i
end for

while a < b
    a = a + 5
end while

repeat
    x = x − 3
while x > 0

In DİM flowcharts a counter loop is also drawn with a diamond: first i = 1, then the diamond i ≤ n, and the body must contain i = i + 1. As soon as you see an arrow going back up, you know there is a loop; look at which exit of the diamond goes back — that is the condition for continuing the loop.

Trace tables and typical loop tasks

In a loop’s trace table each row is one check of the condition: we write the answer to the condition, then the new values of the variables after the body. In the last row the condition is “No” and the loop ends. Note: the condition is always checked one time more than the number of iterations.

n = 4072
s = 0
k = 0
while n > 0:
    s = s + n % 10
    k = k + 1
    n = n // 10
print(s, k)

a = [12, 7, 25, 3, 18]
m = a[0]
for x in a:
    if x > m:
        m = x
print(m)

13 4
25

DİM tasks: the number of iterations and two consecutive loops

In DİM loop tasks the loop often repeats dozens of times — writing every step is long and risky. So you need the pattern: if a variable changes by the same amount on every pass, its value after k passes can be written with a formula. Then the stopping condition of the loop becomes an inequality.

1. 1
   Find the variables

   Which variables are in the loop condition, and by how much do they change on each pass of the body?

2. 2
   Write them after k passes

   For example, a = a₀ + p·k, b = b₀ − q·k; if the change is not constant, write the first 4–5 passes in a table and look for the pattern.

3. 3
   Solve the stopping condition

   Write the opposite of the loop condition (for example, a ≥ b) as an inequality and find the smallest natural k that satisfies it.

4. 4
   Check the boundary

   Compute the values for k − 1 and k: after k − 1 passes the condition must still be true, after k passes it must be false.

Reverse tasks: the input from the outputs

Nested loops and infinite loops

A loop body may contain another loop — this is a nested loop. On every pass of the outer loop the inner loop runs completely from start to end. If the inner loop’s bounds do not depend on the outer variable, the total number of passes is a product: 4 · 3 = 12. If they do (for example, j from 1 to i), you add up the passes row by row.

You have learned to read loops; the next step is to build them yourself. The written part of the exam asks for a complete flowchart that reads n numbers and works with a counter loop — that is the topic of the lesson «Building flowcharts: the written tasks». The Python form of the same loops is studied in the programming section.

a = 3
b = 90
k = 0
# loop here
print(k)

11