Analytical Reasoning & Intelligence Tests — Chapter 6

Other Reasoning: Ranking, Seating, Clocks & Calendar, Classification, Logical Deductions

Why this chapter matters: this is the drawer the banks keep everything that is reasoning but not analogy, series, coding, direction or blood relations — and it is the chapter that closes Subject 12. The bank for it is 18 unique stems across 15 distinct source files: FPSC papers of 2010, 2014 and 2015, PPSC papers of 2019, 2020 and 2021, and CSS papers of 2024 and 2025. No single type dominates; what repeats is the shape of the question — a queue with a position from both ends, a row with two people at the extremes, a clock face, a day-count, a “find the odd one”, a syllogism with two premises. Every stem in this chapter is worked in full below, with its banked answer verified by hand in this run (Section 10). Two stems carry honest flags, not invented logic: the classification stem (Section 6), whose discriminating property the preserved stem does not state, and one number puzzle (Section 7.3), whose wording as preserved does not yield its banked key under the obvious reading. Both are taught as key-plus-flag, never as a made-up rule.

Status of this chapter (read first): this is AR-06, written 31 October 2026 in taxonomy order after AR-05. With it, Subject 12 is complete: AR-01 to AR-06 are all done (Analogies, Series, Coding-Decoding, Direction Sense, Blood Relations, Other Reasoning). The Subject 12 structure page claimed no mark for this family at structure level because its paper-level spread had not been verified file by file; that verification is done in this run (Section 10), and the marks in the Exam Trend box below rest on it — not on the structure page’s counts alone. The script count in this run matches the structure page’s “18 unique stems, 15 files” exactly — no correction needed, unlike AR-04. One classification note, recorded here not hidden: the stem in Section 8 (the anchor blood-relation item under a “Logical Deductions” tag) is counted here because its topic tag puts it in this family, but it is not re-taught — it is the AR-05 anchor stem, and Section 8 simply points back to it, so the chapter teaches 17 new items plus that cross-link.

Exam Trend box

• 🔁 Most Repeated: Other Reasoning as a family — 18 unique stems across 15 distinct source files, clearing the 🔁 paper-count limb (≥5 different papers) three times over: FPSC 2010, 2014, 2015; PPSC 2019, 2020 (×7 files), 2021; CSS 2024, 2025 (×2 files). No single stem repeats as often as AR-05’s anchor, but the types repeat: ranking-from-both-ends appears in three separate papers (Section 2), and the 3:15 clock-angle item appears in two CSS files (Section 4.1).
• ⭐ Must Know: the family as a whole — it is 🔁 and still asked inside the 2024–2026 window: CSS GSA 2024 (the digit-rearrangement puzzle, Section 7.1) and the CSS MPT 2025 General Abilities section (the 3:15 clock angle and the consecutive-even puzzle, Section 4.1 and Section 7.2). Stated honestly: no 2026 filename-year file carries a family-F stem, so the ⭐ claim rests on 2024 and 2025 exam papers, not on a 2026 paper.
• 🆕 Newly Appeared / Recent Trend: none claimed. Every type here runs from FPSC 2010–2015 through PPSC 2019–2021 into CSS 2024–2025. The 2025 items are the same old types (a clock angle, a consecutive-number puzzle), not new types — so this is continuity, recorded as ⭐, not 🆕.

1. The method — five moves, one per type

1. Ranking: add both positions, subtract one. If a person is mth from the left and nth from the right, the total is m + n − 1 — the person is counted twice, once in each count. If a person is mth from either end, the total is 2m − 1. That is the whole method; every ranking stem in the bank is one of these two forms (Section 2).
2. Seating: fix the ends first. “Beside A only” means that person sits at an end, next to A and nobody else. “Fourth from F” fixes the distance between two named people. Place the ends, then fill the middle — never the other way round (Section 3).
3. Clocks: two formulas and one mirror rule. Angle = |30H − 5.5M| degrees; hands coincide 22 times a day, not 24; mirror time = 11:60 − shown time (Section 4).
4. Calendar: count days, not dates. “Day before yesterday” is two days back; “day after tomorrow” is two days forward. Anchor on today and step, never guess (Section 5).
5. Syllogisms: only what the premises force. “All A are B” plus “All B are C” forces “All A are C” — but “All A are B” plus “Some B are C” forces nothing definite about A and C, only a possibility. “May be” is a real answer (Section 8, and the cats stem in Section 7.4).

2. Core notes — Ranking in a queue (3 stems)

2.1 Same person from either end: total = 2m − 1

• FPSC Assistant Director FIA 2015 (topic: Ranking in a queue): “If you are 5th from either end of a queue, how many people does it have?” — 9. 5 + 5 − 1 = 9: you are counted once from the front and once from the back, so subtract the double count.
• FPSC Assistant Security Officer ASF 2014 (topic: Ranking): “If you are 7th in a queue from either end, how many people are there in the queue?” — 13. 7 + 7 − 1 = 13. Same move, different number — this is why the type, not the number, is what you learn.

2.2 Different positions from the two ends: total = m + n − 1

• PPSC ESE General, Labour Department, paper 2, 2019 (topic: Ranking): “In a row, a student is standing 13th from the right and 8th from the left. How many students are there in total?” — 20. 13 + 8 − 1 = 20. The trap answer is 21 (adding without subtracting the double-counted student); the other options (19, 22) are arithmetic slips around the same formula.

3. Core notes — Seating arrangement extremes (1 stem)

• FPSC Assistant Director FIA 2010 (topic: Seating arrangement extremes): “Six persons A, B, C, D, E and F are standing in a row. C and D are standing close to each other alongside E. B is standing beside A only. A is fourth from F. Who are standing on the extremes?” — B and F. Worked: “B beside A only” puts B at an end with A next to it (B at an end has only one neighbour). “A is fourth from F” with six places forces F at the opposite end and A next to B — F, _, _, _, A, B — and C, D and E fill the three middle places, C and D close together alongside E. The extremes are therefore F and B. The trap is placing B in the middle: “beside A only” forbids a second neighbour.

4. Core notes — Clocks (4 stems)

4.1 The angle formula: |30H − 5.5M|

• CSS MPT 2025 General Abilities section (topic: Clock angles): “A clock shows 3:15. The angle between hour and minute hand is:” — 7.5 degrees. |30×3 − 5.5×15| = |90 − 82.5| = 7.5. The trap answers are 30 (the hour hand’s movement in one hour, ignoring the minute hand’s pull), 15 and 40. The same item, worded identically apart from “degrees” for “deg”, also sits in the CSS MPT General Abilities practice-set booklet (topic: Clock angles) with the same key, 7.5 degrees — counted as a second stem by the script because the wording differs by one word, but it is the same item, taught once here.
• Why the formula works, in one line: the hour hand moves 30 degrees per hour (360/12) plus 0.5 degrees per minute, the minute hand moves 6 degrees per minute, so the gap at H hours and M minutes is |30H − 5.5M|.

4.2 Mirror time: 11:60 − shown time

• PPSC Assistant Director S&GAD 2020 (topic: Clocks): “A clock seen through a mirror, shows quarter past three. What is the correct time shown by the clock?” — 8:45. Mirror rule: 11:60 − 3:15 = 8:45. The trap is 9:15 (reflecting the digits instead of the time) and 3:15 itself (forgetting the mirror entirely).

4.3 How often the hands coincide: 22, not 24

• PPSC Ziladar, Irrigation Department, paper 2, 2020 (topic: Clocks): “How many times do the hands of a clock coincide in a day?” — 22. Eleven coincidences per 12 hours (the 11:00–12:00 gap has no coincidence — the hands meet at 12:00, which belongs to the next cycle), doubled for 24 hours. The trap is 24 (once an hour) and 25; the standard answer is 22.

5. Core notes — Calendar logic (1 stem)

• PPSC Junior Clerk S&GAD 2019 (topic: Calendar Logic): “If the day before yesterday was Saturday, then the day after tomorrow will be:” — Wednesday. Step from the anchor: day before yesterday = Saturday, so yesterday = Sunday, today = Monday, tomorrow = Tuesday, day after tomorrow = Wednesday. The trap answers (Thursday, Tuesday, Monday) are all one step off the anchor in one direction or the other — count the steps, don’t guess the day.

6. Core notes — Classification: find the odd one out (1 stem, flagged)

• PPSC Assistant, Local Government, 2020 (topic: Classification): “Find the odd one out from the following:” — options London, Lyon, Sevilla, Belfast — banked answer Belfast. Flag, stated plainly: the preserved stem gives no discriminating property, and no property cleanly separates Belfast from the other three on the evidence in the stem (London and Belfast are both UK capitals in the usual sense; Lyon and Sevilla are non-capital cities, which would make the odd one London, not Belfast). The key is therefore taught as the banked key only — Belfast — with no invented rule attached. If this stem reappears, answer the key; do not spend revision time on a rule the stem does not contain. This is the same honesty standard as AR-01’s five untaught number keys: a key without a reconstructable reason is a flag, not a lesson.

7. Core notes — Number puzzles, anagrams, syllogisms (7 stems)

7.1 Digit rearrangement

• CSS GSA 2024, maths Q31–60 (topic: Number puzzles): “If the digits of the number 4321 are rearranged in ascending order, what is the new number?” — 1234. Ascending means smallest digit first: 1, 2, 3, 4. The trap option 1342 is a partial sort; 4321 is no sort at all. (The banked options carry 1234 twice — a duplicated option in the source file, not a second answer.)

7.2 Consecutive numbers

• CSS MPT 2025 General Abilities section (topic: Number puzzles): “The product of two consecutive even numbers is 48. Find numbers.” — 6, 8. 6 × 8 = 48, and both are even and consecutive. The trap pairs (4, 6) give 24 and (8, 10) give 80. (The banked options carry 6, 8 twice — again a duplicated option, not a second answer.) Method: test the pair around √48 ≈ 7 — the consecutive evens straddling 7 are 6 and 8.

7.3 The flagged number puzzle (key taught, wording flagged)

• PPSC Assistant, Board of Revenue, 2021 (topic: Number Puzzles): “A 2-digit number that is added 3 times and then 45 is added, it reverses the same number. What is the number?” — banked answer 27. Flag, stated plainly: the wording as preserved does not yield 27 under the obvious reading — 3 × 27 + 45 = 126, not 72 — and no two-digit number satisfies 3n + 45 = reverse(n) at all. The key does work under the simpler reading the stem seems to garble: 27 + 45 = 72, the reverse of 27. So the item is taught as: if this stem appears, the banked answer is 27, because 27 + 45 = 72 — and the “added 3 times” phrase is recorded here as a garbled wording, not as a method. Do not practise the 3n + 45 form; it has no solution.

7.4 Syllogisms — what follows, and what only “may be”

• CSS MPT General Abilities practice-set booklet (topic: Logical reasoning): “If all cats are animals and some animals are dogs, then:” — “Some cats may be dogs”. The premises force nothing definite about cats and dogs (the “some animals” that are dogs might or might not include cats), so the definite options (“Some cats are dogs”, “All cats are dogs”, “No cats are dogs”) are all wrong, and the possibility statement is the standard answer. This “may be” move is the most testable syllogism skill in the bank.
• Same booklet (topic: Logical reasoning): “If all roses are flowers and all flowers are plants, then:” — “All roses are plants”. A chain of two “all” premises: roses ⊆ flowers ⊆ plants, so roses ⊆ plants. The trap “All plants are roses” reverses the chain; “Some roses are not plants” contradicts it.

7.5 Reversal code inside a logic topic

• PPSC Junior Clerk, Anti-Corruption, 2020 (topic: Logical Reasoning): “PEACH is to HCAEP then 26541 should be?” — 14562. PEACH reversed is HCAEP, so 26541 reversed is 14562. The topic tag puts this stem in this family, but the method is AR-03’s pure-reversal code — cross-link, don’t re-learn: reverse the whole string, digits included.

7.6 Anagrams — rearrange to a known word, then classify

• PPSC Caretaker, Lahore High Court, 2019 (topic: Anagrams): “If you rearrange the letters ‘RAPIS’, you would get the name of a well-known:” — City. RAPIS rearranges to PARIS, and Paris is a city (the other options: country, ocean, river). Method: try the rearrangement against famous names first — the bank uses real place names, not random strings.

8. Logical deductions — cross-link, not a re-teach

• PPSC Labour Officer 2020 (topic: Logical Deductions): “A is the father of B. But B is not the son of A. Then what is B to A?” — Daughter. This is the AR-05 anchor stem, counted here only because its topic tag (“Logical Deductions”) lands it in family F under the first-match-wins classification. It is taught in full in AR-05 Section 2.1 — the logic is identical (child but not son, therefore daughter) — and is not repeated here beyond this pointer. Counting it twice in two chapters would double-teach one item; the cross-link is the honest treatment.

9. Master table — the whole chapter on one page

10. Rapid-fire one-liners

• Ranking either end: double the position, minus one.
• Ranking two ends: add both positions, minus one.
• “Beside A only” means an end seat — place ends first.
• Clock angle = |30H − 5.5M|; 3:15 gives 7.5 degrees.
• Mirror time = 11:60 − shown time; mirror 3:15 is 8:45.
• Clock hands coincide 22 times a day, not 24.
• Calendar: anchor on today, step two days each way.
• Classification stem (PPSC 2020): banked key Belfast — flagged, no invented rule.
• Ascending 4321 is 1234; consecutive evens for 48 are 6, 8.
• Flagged puzzle: banked key 27, because 27 + 45 = 72.
• All–All chains transmit; All–Some gives only “may be”.
• Reversal: PEACH → HCAEP, so 26541 → 14562.
• RAPIS rearranges to PARIS — a city.
• Anchor deduction (AR-05): not his son, but his child — his daughter.
• Data sufficiency (Section 15): judge whether the data answer the question — one statement alone, both together, or neither. OPC method only; no bank stem yet.
• Rank from the last (Section 15A): total − rank + 1 — 13th of 26 is 14th from the last.
• Gaps, not objects (Section 15A): 8 trees at 3 m are 21 m end to end; 5 cars bumper to bumper touch 4 times.
• Number-line distance is absolute (Section 15A): from −1, 10 (11 away) beats −10 (9 away).
• Seat chains (Section 15A): take left/right from the stem’s own anchor pair — 253, 254, 252, 251.
• Comparison chains (Section 15A): “no faster than” is ≤, not = — C ≥ B > A still makes C faster than A.

11. Most-asked types

By type frequency in this chapter’s bank, not by fact — reasoning items repeat by type:

1. Ranking (3 stems, 3 papers) — the most-asked type in this chapter, and the only type with three distinct papers behind it.
2. Clocks (4 stems counting the duplicated 3:15 item, 3 distinct item-types) — angle, mirror, coincidence.
3. Syllogisms + reversal + anagram (4 stems) — the logic half of the chapter, all from CSS practice or PPSC 2019–2020 papers.
4. Everything else (seating, calendar, classification, digit and number puzzles, deduction cross-link) appears once each — learn the method, not the instance.
5. Data sufficiency: zero bank stems (Section 15) — taught from the OPC book only, so it sits outside this frequency ranking entirely until a paper asks it.
6. OPC pp. 77–79 residues: zero bank stems (Section 15A) — the same OPC-only status: rank from last, gap counting, number-line distance, seat chains and comparison chains are method drills until a paper asks one of them.

12. Traps & confusions

• Ranking: forgetting the −1. 5th from either end is 9, not 10; 13th + 8th is 20, not 21. The double-counted person is the whole trap.
• Seating: putting “beside only” in the middle. A person beside only one other person must be at an end.
• Clock angle: using 30 degrees for 3:15. The hour hand has moved a quarter of the way toward 4 by 3:15; ignoring that gives 30, not 7.5.
• Coincidence: 24 instead of 22. The hands do not meet once an hour — the 11:00 hour has no meeting before 12:00.
• Mirror: reflecting digits instead of time. Mirror 3:15 is 8:45 by the 11:60 rule, not 9:15.
• Calendar: stepping one day instead of two. “Day before yesterday” and “day after tomorrow” are both two steps from today.
• Syllogism: forcing a definite conclusion from All + Some. “Some cats may be dogs” is the answer precisely because nothing definite follows.
• Chain reversal: “All plants are roses”. Chains run one way — roses ⊆ flowers ⊆ plants, never backwards.
• Classification: inventing a rule for Belfast. The stem states none; answer the key and move on (Section 6).
• Flagged puzzle: practising 3n + 45. That form has no two-digit solution; the working form is n + 45 = reverse(n), giving 27 (Section 7.3).
• Data sufficiency: answering from one statement. A > B alone does not make A oldest — C is unplaced (Section 15). The verdict “both together” exists precisely for chains like this; never promote a half-chain to an answer.
• Rank from the last: answering the behind-count. 13th of 26 leaves 13 behind him, but his rank from the last is 14 — count himself (Section 15A).
• Gap counting: multiplying by n. 8 trees are 7 gaps (21 m, not 24); 5 cars are 4 joins, not 5 (Section 15A).
• Number-line distance: picking by sign. From −1, −10 is 9 away and 10 is 11 away — distance ignores sign (Section 15A).
• Seat chains: assuming right = +1. Take the direction from the stem’s anchor pair; another paper may number the other way (Section 15A).
• Comparison chains: “no faster than” as equality. It means ≤, not = — and the conclusion survives either way when the chain is C ≥ B > A (Section 15A).
• The OPC heading trap: “Direction Sense” pp. 77–79 contains no directions. File these items under ranking/counting (this chapter), never under AR-04.

13. Memory tricks

• Ranking rhyme (procedural, not a bank fact): “Both ends count you twice — add, then take one away.” Say it before every ranking stem and the −1 never gets lost.
• Clock-angle habit: always write 30H first, then 5.5M, then subtract — the two numbers (90 and 82.5 for 3:15) make the 7.5 obvious and stop the 30-degree trap.
• Mirror habit: write “11:60” above every mirror-clock stem before you touch the shown time.
• Syllogism habit: draw two nested circles for All–All (transmission is visible) and overlapping circles for All–Some (the overlap may or may not touch A — hence “may be”).
• Sufficiency habit: before solving, ask “can this data even answer it?” — write S1?, S2?, S1+S2?, and answer the sufficiency question first (Section 15).
• Gap rhyme (procedural, not a bank fact): “Objects minus one gives the gaps.” Say it before every trees/cars/poles stem (Section 15A).
• Rank-from-last habit: write total − rank + 1 before you touch the behind-count (Section 15A).
• No invented mnemonics are presented as bank facts in this chapter: the tricks above are procedures for applying the methods, and the flagged stems (Section 6, Section 7.3) carry no trick at all — a trick on a garbled stem would teach the garble.

14. Visuals (declared specs only — site build renders these)

1. Ranking number-line diagram — a queue strip with a person marked m-th from the left and n-th from the right, the double-counted person highlighted, placed under Section 2.
2. Seating row diagram — six slots F–––_–A–B with C, D, E in the middle, placed under Section 3.
3. Clock-face diagrams — 3:15 with the 7.5-degree gap marked, and a mirror-clock pair (3:15 / 8:45), placed under Section 4.
4. Syllogism circle diagrams — nested circles (roses–flowers–plants) and overlapping circles (cats–animals–dogs), placed under Section 7.4.
5. Master-table card — the Section 9 table as a single revision card, placed at the chapter end.
6. Data-sufficiency verdict strip — the four verdicts (S1 alone / S2 alone / both together / neither) with the A > B > C chain diagram, placed under Section 15.
7. Gap-counting strip — 8 trees with 7 gaps marked (21 m) beside 5 cars with 4 joins marked, placed under Section 15A.
8. Seat-number strip — seats 251–254 with Henry, Marla, George and Inez marked, placed under Section 15A.

15. Data sufficiency — a type the banks have not asked (OPC method only)

First, the distinction the OPC book draws (Maths p. 54), which no AR chapter had stated: logical reasoning works on patterns and relationships (AR-01–AR-05: analogies, series, codes, directions, blood relations), while analytical reasoning arranges, compares and evaluates given data (this chapter: ranking, seating, clocks, calendars, deductions). Data sufficiency sits on the analytical side — it asks not for the answer, but whether the data can give the answer.

The question form. A question is given with two statements, and you must judge sufficiency, not solve:

• Statement 1 alone is sufficient, or
• Statement 2 alone is sufficient, or
• Both statements together are sufficient (neither alone is), or
• Neither statement, even together, is sufficient.

OPC worked example (Maths p. 56), recomputed in this run: “Statement 1: A is older than B. Statement 2: B is older than C. Question: Who is oldest?” — A, and the data are sufficient only together. Statement 1 alone gives A > B but says nothing about C; Statement 2 alone gives B > C but says nothing about A. Together they chain to A > B > C, so A is oldest. The verdict is “both together”, not “Statement 1” — answering from one statement alone is the whole trap of the type.

Why it is taught with no mark and no bank count: a text-pattern scan over the same bank dirs as this chapter’s count (script hidden_files/goldmine-notes-count-ar-opc-logic.py, 608 files / 58,098 questions) found zero data-sufficiency stems — no “sufficient to answer”, no “data sufficiency”, no “data is sufficient” wording. It is filed here because the OPC base book teaches it as part of analytical reasoning and one-paper syllabi name the type, not because any paper in our banks has asked it. If a future paper does, this section gains its first bank stem and the front-matter is revisited then; until then, no 🔁, ⭐ or 🆕 is claimed, and the practice bankCount below is unchanged at 18.

15A. The OPC “Direction Sense” block (Maths pp. 77–79) — ranking and counting, not directions

Read the heading with suspicion. The One Paper Cracker Maths book labels its pp. 77–79 block “DIRECTION SENSE”, but not one of its ten items uses a compass direction, a turn, or a walk. Every item is ranking, counting, or comparison — the families of this chapter, not of AR-04. Genuine direction items (walks, turns, relative position) live in AR-04 and nowhere else; filing these ten items under “direction sense” would misfile the whole block. Three of the ten are already taught here and are not re-taught; the other seven add five small sub-types and one scaling line, all OPC drill, none banked (see the evidence paragraph at the end of this section).

Already taught — checked, not duplicated. Item 1, “a student standing 13th from the right and 8th from the left”, is Section 2.2’s PPSC 2019 stem verbatim (13 + 8 − 1 = 20). Item 5, Nyla 9th from both ends, is the Section 2.1 rule (8 + 8 + 1 = 17). Item 7, Asif 6th from either side, is the same rule (5 + 1 + 5 = 11). Same methods, different numbers — the type is what you learn, and these three numbers add nothing new.

New sub-type 1 — rank from the last: total − rank + 1. OPC item 2: “A boy ranks thirteen in a class of twenty-six. What is his rank from the last?” — 14th from the last. 26 − 13 + 1 = 14. OPC’s own working agrees (13 students behind him, so he is 14th counting himself). The trap is answering 13 — the behind-count — instead of the rank, which counts the boy himself. This is the mirror image of Section 2: Section 2 finds the total from two positions; this finds one position from the total and the other position.

New sub-type 2 — gap counting: n objects have n − 1 gaps. OPC item 3: 8 mango trees in a straight line, 3 metres between each tree — the distance from the first to the eighth tree is (8 − 1) × 3 = 21 metres, not 8 × 3 = 24. OPC item 9 is the same rule wearing different clothes: 5 cars parked bumper to bumper have 4 bumpers touching, because 5 cars make 4 joins; the front bumper of the first car and the back bumper of the last stay free. Trees, cars, poles, fence posts — whenever objects stand in a line, count the gaps, not the objects. The trap in both faces is the same: multiplying by n instead of n − 1.

New sub-type 3 — distance on a number line is absolute. OPC item 4: “Find the farthest number from −1” among −10, 10, 5, 1 — 10. Distances from −1 are |−10 − (−1)| = 9, |10 − (−1)| = 11, |5 − (−1)| = 6, |1 − (−1)| = 2. The trap is picking −10: it looks “farthest” by sign or by size on the left, but distance ignores sign, and 11 beats 9.

New sub-type 4 — a seat-number chain. OPC item 6: at the baseball game, Henry sits in seat 253, Marla to his right in seat 254, George to Henry’s left, Inez to George’s left — Inez sits in seat 251. George is 252 (one below Henry), Inez is 251 (one below George). The flag on this item matters more than its arithmetic: “right = +1” is this stem’s own convention, read off its anchor pair (Henry 253, Marla to his right 254). Another paper may number seats the other way; never assume right means +1 — always take the direction from the stem’s first anchored pair, then step.

New sub-type 5 — a comparison chain. OPC item 10: “A runs less fast than B, and B runs fast but no faster than C. As compared to A, C runs…” — faster than A. The chain is A < B and B ≤ C, so C ≥ B > A. One correction to OPC’s own working, recorded not hidden: OPC writes “B is as fast as C”, but “no faster than” means B ≤ C, not B = C — B could be slower than C and the conclusion still holds, because C ≥ B already beats A. The trap is answering from A < B alone, or treating “no faster than” as equality and then doubting the conclusion when equality fails.

Scaling, filed for completeness. OPC item 8: four people live in a house, 22 houses in one street — 22 × 4 = 88 people in the street. Pure multiplication, no method to learn; it is filed here so the block is complete, not because it teaches a type.

Bank evidence — why nothing here carries a mark. A tightened text-pattern scan over the same bank dirs as this chapter’s count (script hidden_files/goldmine-notes-count-ar06-opc77.py, 608 files / 58,098 questions) finds zero genuine stems for rank-from-last, gap-trees, bumpers, number-line distance, seat-number chains, “no faster than” comparisons, and house-scaling alike. A loose first pass returned false positives that are excluded, not counted: geography “poles” hits for the trees pattern, “farthest planet” hits for the number-line pattern, and a series item’s “fourth term from last” for the rank pattern. This section is therefore OPC method drill only: no 🔁, no ⭐, no 🆕, and the practice bankCount below is unchanged at 18. If a future paper asks any of these sub-types, this section gains its first bank stem and the front-matter is revisited then.

16. Practice this topic

Pull a timed set from the verified banks using the practice query in the front-matter (Ranking, Seating arrangement, Clock angles, Clocks, Calendar Logic, Classification, Anagrams, Number puzzles, Logical reasoning, Logical Deductions tags): 18 unique stems across 15 distinct files. The Section 15A OPC pp. 77–79 residues are drill material from the OPC book, not banked practice: no bank stem is added by Section 15A, and the count stays 18. Book mode: ← Previous: Blood Relations (AR-05) · Next: GK Economics & Finance Basics (next coverage-audit item, to follow) →

Sources & count (Section 16 honesty note). Count by script hidden_files/goldmine-notes-count-ar06.py (counts JSON hidden_files/goldmine-notes-ar06-counts.json), same bank dirs and topic-based classification order (A→F, first match wins) as the Subject 12 structure count (hidden_files/goldmine-notes-count-ar12.py): 586 files / 56,529 questions scanned; 18 rows including duplicates; 18 unique stems across 15 distinct files; filename-year files 2024: 1, 2025: 1, 2026: none. This matches the structure page’s “18 unique stems, 15 files” exactly — no correction needed, unlike AR-04. Stem sources: FPSC Assistant Director FIA 2010 (seating, 1), FPSC Assistant Security Officer ASF 2014 (ranking, 1), FPSC Assistant Director FIA 2015 (ranking, 1), PPSC Caretaker Lahore High Court 2019 (anagram, 1), PPSC ESE General Labour Department paper 2 2019 (ranking, 1), PPSC Junior Clerk S&GAD 2019 (calendar, 1), PPSC Assistant Director S&GAD 2020 (mirror clock, 1), PPSC Assistant Local Government 2020 (classification, 1), PPSC Junior Clerk Anti-Corruption 2020 (reversal, 1), PPSC Labour Officer 2020 (logical deduction cross-link, 1), PPSC Ziladar Irrigation Department paper 2 2020 (coincidence, 1), PPSC Assistant Board of Revenue 2021 (flagged number puzzle, 1), CSS GSA 2024 maths Q31–60 (digit rearrangement, 1), CSS MPT 2025 General Abilities section (clock angle + consecutive evens, 2), CSS MPT General Abilities practice-set booklet (clock angle duplicate + 2 syllogisms, 3). Every banked answer taught here was verified by hand against its stem in this run: ranking 9 / 13 / 20 by the m + n − 1 rule; seating extremes B and F by the F–––_–A–B placement; clock angle 7.5 by |30×3 − 5.5×15|; mirror 8:45 by 11:60 − 3:15; coincidence 22 (11 per 12 hours); calendar Wednesday by two-day stepping from Saturday; digit sort 1234; consecutive evens 6 × 8 = 48; syllogisms “may be” and “All roses are plants” by premise logic; reversal 14562; anagram PARIS → City; anchor deduction Daughter (AR-05 Section 2.1). Two stems are taught as key-plus-flag, not as rules: the classification key Belfast (Section 6 — no discriminating property stated in the stem) and the Board of Revenue number key 27 (Section 7.3 — the preserved “added 3 times” wording yields no solution; the working reading is 27 + 45 = 72). Duplicated banked options (1234 twice; 6, 8 twice) are source-file duplicates, not second answers. No fact in this chapter comes from memory; methods are standard procedural types, and every worked example is a bank stem with its banked answer. OPC extension (this run, 2 November 2026): the One Paper Cracker Maths logical/analytical block (pp. 50–64, OCR hidden_files/final-notes-intake/work/maths-opc-pp51-60.txt and maths-opc-pp61-70.txt, 20/20 pages clean) was checked against this chapter and AR-01–AR-05. Its only addition here is Section 15 Data sufficiency (p. 56) plus the p. 54 logical-vs-analytical distinction that opens it — both OPC-sourced, zero bank stems by the scan in Section 15, no mark claimed, bankCount unchanged at 18. The rest of the block is checked-duplicate: number/letter series → AR-02, analogies (a)–(i) → AR-01 (gender = its Female : male row), coding → AR-03, direction → AR-04, blood relations → AR-05, ranking/seating/comparison/order → Sections 2–3 above, statement-and-conclusion and Venn circles → Section 7.4/Section 13’s method, and the cause-and-effect question type is held, not taught — zero genuine bank stems, a trivial example, and the cause : effect relation already lives in AR-01 Section 3.5. The block’s Time & Work / Distance & Speed / Age pages (pp. 64–70+) are separate later OPC units, not this one. OPC extension (this run, 2 November 2026): the pp. 77–79 “Direction Sense” block (OCR hidden_files/final-notes-intake/work/maths-opc-pp71-80.txt, pp. 77–79) was checked against this chapter and AR-04. Its addition here is Section 15A — rank from last, gap counting, number-line distance, seat-number chains, comparison chains, plus the 22 × 4 scaling line — with three items checked-duplicate (Section 2.1/Section 2.2 ranking) and OPC’s “B = C” reading corrected to B ≤ C. Tightened text-pattern scan (script hidden_files/goldmine-notes-count-ar06-opc77.py, 608 files / 58,098 questions): zero genuine bank stems for all seven patterns, so no mark claimed and bankCount unchanged at 18. Next OPC unit: Basic Statistics (pp. 80–81) — p. 80 is OCR’d in the same file, p. 81 still un-OCR’d at this run.