Sequential Output Tracing

What you need to understand

The rule is stated in full, so only two things can go wrong: what each step does, and how many steps have been taken.

Each step settles one number in its final position, so the number of steps equals the number of numbers.

Only the numbers not yet placed are considered when looking for the largest, which is what makes the process move forward.

The place the largest number moves to is fixed in advance: the leftmost place not yet filled.

Numbers already placed never move again, so the line is partly settled as you go.

Write the line out after each step rather than trying to hold several steps in your head.

How to work through these questions

Write the input line down exactly as given.

Apply one step at a time, writing out the whole line after each and numbering the step.

Mark the places already filled so you do not disturb them.

Stop at the step the question asks about, not at the end of the process.

Check how many steps you have taken, since one step too few or too many is the usual error.

If the question asks how many steps, remember that the final step may already leave the line in order.

Mistakes that cost marks

Reporting the line one step early or one step late.
Moving a number that has already been placed in its final position.
Choosing the largest number anywhere in the line rather than the largest among those not yet placed.
Giving the finished line when the question asked for a line partway through.
Counting the steps from zero instead of from one.
Forgetting that a step which leaves the line unchanged still counts as a step.

Worked example

A machine rearranges a line of numbers. At each step it finds the largest number not yet placed in its final position and swaps it with the leftmost number not yet placed. Input: 23 61 20 94 93 19. What is the line after step 2?

The largest number is 94 and the leftmost place is the first, so swapping gives 94 61 20 23 93 19 after step one.
The first place is now settled, so the largest of the remaining numbers is 93, and the leftmost free place is the second.
Swapping 93 into the second place gives 94 93 20 23 61 19.
The line after step 2 is therefore 94 93 20 23 61 19.

Answer: 94 93 20 23 61 19

Practice questions with answers

A few Sequential Output Tracing questions with the full solution shown, so you can see how the method is applied before you attempt the timed set.

Question 1

A machine rearranges a line of numbers. At each step it looks at the numbers not yet placed in their final positions, finds the largest of them, and swaps it with the leftmost number not yet placed. Step 1 puts the largest number in the first position, step 2 puts the next largest in the second position, and so on until the line is in descending order. Input: 23 61 20 94 93 19 56 What is the line after step 4?

A 94 93 61 56 23 19 20
B 94 93 61 56 23 20 19
C 94 93 61 23 20 19 56
D 94 93 61 56 20 19 23

Answer: Option D — with explanation

Working through the rule: step 1 places 94, the largest number, in the first position; each later step does the same for what is left. After step 4 the line reads 94 93 61 56 20 19 23. Common mistakes - reported the line one step too far on: 94 93 61 56 23 19 20 - gave the finished line instead of the line after that step: 94 93 61 56 23 20 19 - reported the line one step too early: 94 93 61 23 20 19 56

Question 2

A 95 70 45 43 41 27
B 95 43 27 41 45 70
C 95 70 27 41 45 43
D 95 70 45 41 27 43

Answer: Option C — with explanation

Working through the rule: step 1 places 95, the largest number, in the first position; each later step does the same for what is left. After step 2 the line reads 95 70 27 41 45 43. Common mistakes - gave the finished line instead of the line after that step: 95 70 45 43 41 27 - reported the line one step too far on: 95 70 45 41 27 43 - reported the line one step too early: 95 43 27 41 45 70

Question 3

A 79 73 58 65 15 71
B 79 73 71 65 15 58
C 79 58 73 65 15 71
D 79 73 71 65 58 15

Answer: Option A — with explanation

Working through the rule: step 1 places 79, the largest number, in the first position; each later step does the same for what is left. After step 2 the line reads 79 73 58 65 15 71. Common mistakes - reported the line one step too far on: 79 73 71 65 15 58 - reported the line one step too early: 79 58 73 65 15 71 - gave the finished line instead of the line after that step: 79 73 71 65 58 15

Question 4

A machine rearranges a line of numbers. At each step it looks at the numbers not yet placed in their final positions, finds the largest of them, and swaps it with the leftmost number not yet placed. Step 1 puts the largest number in the first position, step 2 puts the next largest in the second position, and so on until the line is in descending order. Input: 36 73 60 81 30 88 37 What is the line after step 2?

A 88 81 73 60 37 36 30
B 88 81 60 73 30 36 37
C 88 81 73 60 30 36 37
D 88 73 60 81 30 36 37

Answer: Option B — with explanation

Working through the rule: step 1 places 88, the largest number, in the first position; each later step does the same for what is left. After step 2 the line reads 88 81 60 73 30 36 37. Common mistakes - reported the line one step too far on: 88 81 73 60 30 36 37 - gave the finished line instead of the line after that step: 88 81 73 60 37 36 30 - reported the line one step too early: 88 73 60 81 30 36 37

Question 5

A machine rearranges a line of numbers. At each step it looks at the numbers not yet placed in their final positions, finds the largest of them, and swaps it with the leftmost number not yet placed. Step 1 puts the largest number in the first position, step 2 puts the next largest in the second position, and so on until the line is in descending order. Input: 45 85 19 38 70 32 55 What is the line after step 2?

A 85 70 55 45 38 32 19
B 85 45 19 38 70 32 55
C 85 70 55 38 45 32 19
D 85 70 19 38 45 32 55

Answer: Option D — with explanation

Working through the rule: step 1 places 85, the largest number, in the first position; each later step does the same for what is left. After step 2 the line reads 85 70 19 38 45 32 55. Common mistakes - reported the line one step too early: 85 45 19 38 70 32 55 - gave the finished line instead of the line after that step: 85 70 55 45 38 32 19 - reported the line one step too far on: 85 70 55 38 45 32 19

Frequently asked questions

Do I need to work through the whole process?

No. Stop at the step the question asks about. Working further costs time and makes it likelier that you report the wrong line.

Why does the largest number go to the leftmost free place?

Because that is what the stated rule says. These questions are about following the rule, not about choosing a sensible one.

What if the largest number is already where it should move to?

Then nothing changes at that step, but it still counts as a step, and the next step moves on to the next largest number.

How many steps does the process take in total?

One step for each number, since each step fixes one number in its final position.

Is it worth writing out every step?

Yes. Writing the whole line after each step is a little slower for the first step and much faster overall, because it removes the need to hold the line in your head.