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Reasoning · Chapter 9

Sequence Counting

Long strings of letters, numbers and symbols with a "how many are preceded by… and followed by…" question attached. Two or three marks in every 2025 shift, and pure procedure once you stop reading and start scanning.

Concept video — watch on the AAI ATC Exam Prep channel
2–3questions per shift
60 sectarget time each
3sequence types
6verified PYQs inside
Objectives 1. The scan method 2. Letter sequences 3. Number sequences 4. Symbol sequences 5. The traps Summary Assessment Free PDF
Before you start

Learning objectives and outcomes

There is no cleverness in this chapter at all. There is a procedure, and candidates who follow it score; candidates who eyeball the string do not.

Objective 1Read the condition first and write it as a three-slot pattern before looking at the sequence.
Objective 2Scan the string once, left to right, testing each element against the middle slot.
Objective 3Apply vowel, consonant, odd, even, square and symbol conditions without hesitating.
Objective 4Handle the first and last elements correctly, since they can never satisfy a two-sided condition.
Outcome AFinish a 27-element string in about sixty seconds with a reliable count.
Outcome BNotice when "and" makes a condition two-sided rather than one-sided.
Subtopic 1

The three-slot scan

Every one of these questions describes a window of three consecutive elements: something before, the thing you are counting, and something after. Write that window down before you read the string.

Turn the sentence into a pattern

"How many symbols are immediately preceded by a letter
and immediately followed by another symbol?"

becomes   [ letter ] [ symbol ] [ symbol ]

Now you are not solving a puzzle, you are matching a three-character pattern across a string. Slide the window one place at a time and tick every match.

The rules that never change

Counting always runs left to right only — every AAI question states this explicitly. The first element can never qualify if the condition needs something before it, and the last element can never qualify if the condition needs something after it. Those two exclusions alone remove the most common wrong answer.

Figure 9.1 — The sliding three-element window A string of letters and symbols with a rectangular window drawn over three consecutive elements, and arrows showing it sliding one place to the right. Upload image here
✈ AAI ATC relevance — the 2025 Shift 1 paper carried three of these, Shift 2 carried two, and the 2021 papers carried one each
Memory trick — "Pattern first, string second"

Write the three-slot pattern in the margin before your eyes go anywhere near the sequence. Reading the string first and the condition second is how people end up counting the wrong thing.

EasyATC Pattern
A question asks how many consonants are immediately preceded by a vowel and immediately followed by a vowel. Which three-slot pattern does this describe?
The thing being counted always sits in the middle slot. Here that is the consonant, with a vowel on each side. Writing this down first is the entire technique.
ModerateATC Pattern
In a sequence of 20 elements, how many positions could possibly satisfy a condition that requires both a preceding and a following element?
The first element has nothing before it and the last has nothing after it, so both are excluded. That leaves positions 2 through 19, which is 18 positions. Knowing this stops you wasting time on the two ends.
HardATC Pattern
A condition reads "immediately preceded by a letter and also immediately followed by another symbol". What does the word "another" tell you?
"Another symbol" implies the element in the middle slot is already a symbol, so the following element must be a second one. The word carries real information about the middle slot, which is why the condition should be read word by word rather than skimmed.
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Subtopic 2

Letter sequences — vowels and consonants

The string is pure letters and the condition is built on vowels and consonants. Mark every vowel before you start scanning; the consonants are then everything you did not mark.

Mark the vowels first

Vowels are A, E, I, O and U. Y is treated as a consonant in these questions.

Underline or circle each vowel in one quick pass. The scan afterwards becomes a matter of looking at marks rather than reading letters, which is roughly twice as fast.

Worked example — 2025 Shift 1 paper

How many consonants are there in the following series, each of which is immediately preceded by a vowel and also immediately followed by a vowel?

(LEFT) S I N G L E R O F I H T P Q D U B W N M J A K E V G P (RIGHT)
  1. Pattern: [vowel] [consonant] [vowel].
  2. Mark the vowels: I at position 2, E at 6, O at 8, I at 10, U at 16, A at 22, E at 24.
  3. Now look only at the gaps between two marked vowels that hold exactly one letter.
  4. Between E and O sits R. Between O and I sits F. Between A and E sits K.
  5. Check the near misses: N at position 3 has a vowel before it but G after it. H at 11 has I before but T after. V at 25 has E before but G after.
  6. Answer: Three.
Figure 9.2 — Vowels marked, gaps inspected The letter string written out with every vowel circled, and the three single-letter gaps between circled vowels highlighted. Upload image here
Memory trick — "Vowel, gap, vowel"

Once the vowels are circled, you are looking for two circles with exactly one letter between them. Nothing else on the page matters.

EasyATC Pattern
In the series A B E C I D O, how many consonants are immediately preceded by a vowel and immediately followed by a vowel?
The vowels sit at positions 1, 3, 5 and 7. That leaves B between A and E, C between E and I, and D between I and O — three consonants, each with a vowel on both sides.
ModeratePYQ 2025 · Shift 1
Refer to the series below. Counting is from left to right only. How many such consonants are there, each of which is immediately preceded by a vowel and also immediately followed by a vowel? S I N G L E R O F I H T P Q D U B W N M J A K E V G P
The qualifying consonants are R, sitting between E and O; F, sitting between O and I; and K, sitting between A and E. Three in total. Anyone who counts N at the start — which has I before it but G after — lands on option 3.
HardPYQ 2021 · Shift 2
How many letters are there in the following series that are not vowels and are immediately preceded by an even number? D 22 A G 3 B 62 O 2 N 7 M 12 L Q K 10 E 43 T 23 F 5
Only two qualify: N, preceded by 2, and L, preceded by 12. The near misses are deliberate — A is preceded by 22 and O by 62, but both are vowels, and E is preceded by 10 but is also a vowel. This condition is one-sided, so nothing after the letter matters at all.
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Subtopic 3

Number sequences — odd, even and squares

The string is single digits and the condition uses odd, even, prime or perfect square. AAI adds a bracketed note when a definition might be disputed — in 2025 it printed "1 is a perfect square" so nobody could argue.

The single-digit facts you need

Perfect squares: 1, 4, 9  ·  Perfect cubes: 1, 8
Primes: 2, 3, 5, 7  ·  Even: 2, 4, 6, 8  ·  Odd: 1, 3, 5, 7, 9

Note that 0 is usually absent from these strings, and that 1 counts as a perfect square but not as a prime. Both of those are examiner favourites.

Worked example — 2025 Shift 1 paper

How many perfect squares are there, each of which is immediately preceded by an even digit and immediately followed by an odd digit? The paper notes that 1 is a perfect square.

(LEFT) 1 4 2 3 9 4 8 1 5 7 9 7 2 5 1 3 4 6 7 8 2 9 5 6 3 6 8 (RIGHT)
  1. Pattern: [even] [square] [odd], where square means 1, 4 or 9.
  2. Scan for squares. The 1 at position 8 has 8 before it and 5 after it — even then odd. That qualifies.
  3. The 9 at position 22 has 2 before it and 5 after it — even then odd. That qualifies too.
  4. Check the rest: the 4 at position 2 is preceded by 1, which is odd. The 9 at position 5 is preceded by 3. The 4 at position 6 is preceded by 9. The 1 at position 15 is preceded by 5. The 4 at position 17 is preceded by 3. All fail on the first slot.
  5. Answer: Two.
Figure 9.3 — Scanning a digit string The digit string written out with the three-slot window highlighted at each of the two qualifying positions, and the failed candidates crossed through. Upload image here
Memory trick — "Squares are 1, 4, 9 — nothing else"

In a single-digit string there are only three perfect squares, so start by circling every 1, 4 and 9. There are usually six or seven of them, and each takes two seconds to test.

EasyATC Pattern
In the string 2 4 7 6 9 3 8 1 5, how many odd digits are immediately preceded by an even digit?
The 7 is preceded by 4, the 9 by 6 and the 1 by 8 — three in total. The 3 is preceded by 9 and the 5 by 1, both odd, so those fail. The first digit is even and is excluded anyway.
ModeratePYQ 2025 · Shift 1
Counting from left to right only, and taking 1 as a perfect square, how many perfect squares are immediately preceded by an even digit and immediately followed by an odd digit? 1 4 2 3 9 4 8 1 5 7 9 7 2 5 1 3 4 6 7 8 2 9 5 6 3 6 8
Two qualify — the 1 sitting between 8 and 5, and the 9 sitting between 2 and 5. Every other square in the string fails because the digit before it is odd. The bracketed note about 1 exists precisely so that the first of these two counts.
HardATC Pattern
In the string 3 8 5 2 7 4 9 6 1 8 3, how many prime digits are immediately preceded by an even digit and immediately followed by an even digit?
The primes are 3, 5, 2, 7 and 3. The 5 sits between 8 and 2, both even, so it qualifies. The 7 sits between 2 and 4, both even, so it qualifies too. The 2 at position 4 has 5 before it, the first 3 has nothing before it, and the last 3 has nothing after it. That gives two.
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Subtopic 4 · highest yield

Symbol sequences

Both 2025 shifts carried a symbol question, and both used the identical condition: symbols preceded by something and followed by another symbol. The trick is that consecutive symbols disqualify each other on the left.

Only the first symbol of a run can qualify

If the condition needs a letter or a number immediately before, then inside a run of three symbols only the first one has a non-symbol before it. The second and third are automatically out. That observation kills most candidates in one line.

L # ?  →  the # qualifies, the ? cannot
because ? has a symbol before it, not a letter
Worked example — 2025 Shift 2 paper

How many such symbols are there, each of which is immediately preceded by a letter and also immediately followed by another symbol?

(LEFT) N + @ # A ₹ M S T E % R & A K G W < O = P Z D ^ U * Q B F (RIGHT)
  1. Pattern: [letter] [symbol] [symbol].
  2. Find every run of two or more symbols, then check whether a letter sits immediately before the run.
  3. The only run of two or more here is the + @ # block near the start, and it is preceded by the letter N. So the + qualifies.
  4. The @ and the # inside that run have symbols before them, so they fail the first slot.
  5. Every other symbol in the string — the ₹, %, &, <, =, ^ and * — sits alone between two letters, so each fails the third slot.
  6. Answer: One.
Figure 9.4 — Runs of symbols The symbol string with each run of consecutive symbols boxed, and only the first symbol of each run marked as a candidate. Upload image here
Memory trick — "Find the runs, then check the left"

Do not scan symbol by symbol. Scan for places where two symbols sit side by side, because the condition demands one. Then check what comes before each run. In a 29-element string this reduces the work to two or three checks.

EasyATC Pattern
In a run of four consecutive symbols preceded by a letter, how many of those symbols are immediately preceded by a letter?
Only the first symbol in the run has the letter immediately before it. The other three each have a symbol before them. This is why long runs of symbols contribute almost nothing to the count.
ModeratePYQ 2025 · Shift 2
Counting from left to right only, how many such symbols are there, each of which is immediately preceded by a letter and also immediately followed by another symbol? N + @ # A ₹ M S T E % R & A K G W < O = P Z D ^ U * Q B F
Only the + qualifies, sitting between the letter N and the symbol @. Every other symbol in this string is isolated between two letters, so none of them has another symbol following it. The string looks crowded with symbols, which is exactly the point.
HardPYQ 2025 · Shift 2
All numbers are single digits. How many such symbols are there, each of which is immediately preceded by a number and immediately followed by another symbol? 2 # ! 3 4 @ * $ 5 1 = + > 8 9 & $
Four runs of symbols each begin right after a digit: the # after 2, the @ after 4, the = after 1, and the & after 9. In each case the next element is also a symbol, so all four qualify. The symbols deeper inside each run — the !, the * and $, the + and >, and the final $ — all fail because a symbol precedes them.
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Subtopic 5

The five traps

These questions are not difficult, so the examiner builds the difficulty into the wording. Five traps account for almost every wrong answer.

What to watch for

  1. One-sided versus two-sided. "Preceded by an even number" needs only the left. Adding "and followed by" doubles the work and usually halves the count.
  2. The end elements. The first and last can never satisfy a two-sided condition. Do not test them.
  3. The word "another". It tells you the middle element is already of that type.
  4. Left to right only. Every paper states it. It means you never count a pair backwards.
  5. The bracketed note. When the paper says 1 is a perfect square, that note exists because it changes the answer.
Figure 9.5 — One-sided versus two-sided conditions The same string shown twice, once with a one-sided condition producing four hits and once with the two-sided version producing two. Upload image here
Memory trick — "Circle the word AND"

Before anything else, circle the word "and" in the question if it is there. It is the single word that decides whether you check one neighbour or two.

EasyATC Pattern
Which element of a sequence can never satisfy a condition requiring something immediately after it?
The last element has nothing following it, so a condition needing a follower can never be met there. Symmetrically, the first element fails any condition needing a predecessor.
ModeratePYQ 2025 · Shift 1
How many symbols are immediately preceded by a letter and immediately followed by another symbol? L # ? M ! / + O K @ N & A % E * $ J = P {
Three qualify. The # sits after the letter L and before the symbol ?. The ! sits after M and before /. The * sits after E and before $. The @, &, % and = each sit between two letters, and the closing brace has nothing after it at all.
HardATC Pattern
Using the same string, how many symbols are immediately preceded by a letter, ignoring what follows them? L # ? M ! / + O K @ N & A % E * $ J = P {
Dropping the second half of the condition, the qualifying symbols are #, !, @, &, %, *, = and the closing brace — eight in all. Note that the brace now counts, because nothing needs to follow it. Compare eight with the three from the previous question and you can see exactly how much work the word "and" was doing.
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Exam strategy

Six tips that hold in the AAI ATC hall

Write the pattern firstThree slots in the margin, before you look at the string.
Circle the word "and"It decides whether you check one neighbour or two.
Skip the two endsThey cannot qualify for a two-sided condition.
Mark the rare type firstVowels, squares or runs of symbols — whichever the condition centres on.
Only the first of a runIn consecutive symbols, only the first can have a letter before it.
Read the bracketed noteIf the paper defines something, that definition changes the count.
Quick revision

Chapter summary sheet

The windowEvery condition is a three-slot pattern with the counted item in the middle.
DirectionLeft to right only, always stated in the paper.
End elementsFirst and last never qualify for a two-sided condition.
VowelsA, E, I, O, U. Y counts as a consonant.
Single-digit squares1, 4 and 9 only. AAI notes explicitly that 1 counts.
Single-digit primes2, 3, 5 and 7. One is not prime.
Symbol runsOnly the first symbol of a run can have a letter or digit before it.
2025 answersSquares: two. Consonants: three. Symbols: three, one and four.
2021 answerNon-vowel letters after an even number: two.

↓ Free download — Sequence Counting question PDF

Practice covering letter, number and symbol sequences, with every string printed in wide monospace so you can mark it up directly on the page. Print it, solve it, keep it.

End of chapter

Comprehensive assessment

Multiple choice and assertion–reason, building in difficulty. Full solutions are in the downloadable PDF above.

A · Multiple choice questions

1. Which three-slot pattern describes "consonants preceded by a vowel and followed by a vowel"?
2. In a sequence of 20 elements, how many positions can satisfy a two-sided condition?
3. What does the word "another" tell you about the element being counted?
4. In A B E C I D O, how many consonants have a vowel on both sides?
5. How many consonants have a vowel on both sides?S I N G L E R O F I H T P Q D U B W N M J A K E V G PPYQ 2025 · Shift 1
6. How many non-vowel letters are immediately preceded by an even number?D 22 A G 3 B 62 O 2 N 7 M 12 L Q K 10 E 43 T 23 F 5PYQ 2021 · Shift 2
7. In 2 4 7 6 9 3 8 1 5, how many odd digits are immediately preceded by an even digit?
8. How many perfect squares are preceded by an even digit and followed by an odd digit? Take 1 as a perfect square.1 4 2 3 9 4 8 1 5 7 9 7 2 5 1 3 4 6 7 8 2 9 5 6 3 6 8PYQ 2025 · Shift 1
9. In 3 8 5 2 7 4 9 6 1 8 3, how many primes have an even digit on both sides?
10. In a run of four consecutive symbols preceded by a letter, how many are immediately preceded by a letter?
11. How many symbols are preceded by a letter and followed by another symbol?N + @ # A ₹ M S T E % R & A K G W < O = P Z D ^ U * Q B FPYQ 2025 · Shift 2
12. How many symbols are preceded by a number and followed by another symbol?2 # ! 3 4 @ * $ 5 1 = + > 8 9 & $PYQ 2025 · Shift 2
13. How many symbols are preceded by a letter and followed by another symbol?L # ? M ! / + O K @ N & A % E * $ J = P {PYQ 2025 · Shift 1
14. Using the same string, how many symbols are preceded by a letter, ignoring what follows?
15. Which element can never satisfy a condition requiring something immediately after it?

B · Assertion and reason

16. Assertion: In a run of consecutive symbols, only the first can be immediately preceded by a letter. Reason: Every symbol after the first has another symbol immediately before it.
17. Assertion: The first element of a sequence can satisfy a condition that requires a preceding element. Reason: Counting is done from left to right only.
18. Assertion: Adding "and immediately followed by" usually reduces the count. Reason: A two-sided condition is satisfied by fewer positions than a one-sided one.
19. Assertion: The 2025 paper's note that 1 is a perfect square was decorative. Reason: No qualifying element in that question was the digit 1.
20. Assertion: Marking every vowel before scanning speeds up a letter-sequence question. Reason: Once vowels are marked, the scan becomes a search for single-letter gaps between marks.
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