Arrangement & Ranking

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This family covers rows, circles, floors and rankings. They look time-consuming, and the ones with eight people can be — but the ranking questions are ten-second marks once you know the counting rule, and arrangement puzzles fall quickly if you start from the most restrictive clue rather than the first one.

The Ranking Rule

Total = (rank from top) + (rank from bottom) − 1

The −1 exists because the person themselves is counted twice. Forgetting it is the single most common error in this topic.

Rearranged: rank from bottom = Total − rank from top + 1

Arrangement Puzzles

Start with the most restrictive clue — the one that fixes a position or allows only a couple of options, not the one printed first.

"Three people between X and Y" means the gap in positions is 4, not 3.

"Immediately below / above" fixes two positions as a single block that can slide together.

Facing matters in circles: for people facing the centre, left is clockwise; for people facing outward, left is anticlockwise. Everything reverses.

Combine parity clues with distance clues first. If two people must sit on even-numbered floors and be four apart, there may be only one possible pair — and the whole puzzle unlocks from there.

Worked Examples from Real Papers

TS Police SI 2022 — Q52

A ranks 13th from the top and B ranks 17th from the bottom in a class. C is 5 ranks below A and 5 ranks above B. How many students are in the class?

Convert everything to ranks from the top.
A is 13th. C is 5 below A → C is 18th. B is 5 below C → B is 23rd from the top.
B is also 17th from the bottom, so Total = 23 + 17 − 1 = 39.
Method: put every person on the same measuring scale before applying the rule. Mixing "from top" and "from bottom" mid-calculation is where this goes wrong.

TS Police Constable 2022 — Q174

In a row of n boys facing north, X is 22nd from the left and Y is 33rd from the right. Z is 3rd to the right of X and 6th to the left of Y. Find n.

X is 22nd from the left, so Z (3 to the right) is 25th from the left.
Y is 6 further right of Z, so Y is 31st from the left.
Y is also 33rd from the right, giving n = 31 + 33 − 1 = 63.

TS Police SI 2022 — Q71

Six persons P₁…P₆ live one on each of six floors, numbered 1 at the bottom to 6 at the top. P₃ lives immediately below P₁. P₅ lives immediately below P₄. Three people live between P₆ and P₅. P₂ is not on the top floor. P₁ does not live on any floor below P₄. P₅ and P₆ live on even-numbered floors. Which floors do P₁ and P₂ occupy?

Start with the most restrictive pair of clues: P₅ and P₆ are both even, and three people between them means their floors differ by 4. Among 2, 4 and 6 the only such pair is 2 and 6.
If P₆ = 2 and P₅ = 6, then P₄ would need floor 7 — impossible.
So P₆ = 6 and P₅ = 2, which forces P₄ = 3.
Floors 1, 4 and 5 remain for P₁, P₂ and P₃. Since P₁ must be above P₄ and P₃ sits immediately below P₁, the pair (P₃, P₁) = (4, 5) is the only fit.
That leaves P₂ on floor 1, which satisfies "not on the top floor".
Answer: P₁ on floor 5 and P₂ on floor 1 — that is, 5 & 1.

"Three people between" is a gap of four. Reading it as three is the most reliable way to lose an arrangement question, because the wrong reading often still produces a self-consistent (but wrong) layout.

Also see: Coding & Decoding · Series · Blood Relations