SPI Reasoning · Order, Ranking and Arrangement

Ranks and Order Puzzles

A rank counts a person's place from one end of a line; an order puzzle builds the whole line from comparisons, then matches each person to a job by cutting what the clues forbid.

All Order, Ranking and Arrangement notes

Why this matters

Five questions in the bank: one EASY rank question from the Sample Paper, and a set of four from the 2022 paper on one puzzle about the incomes and jobs in a family (three MODERATE, one HARD). The puzzle set is won by writing the income line once and answering all four questions from it.

Concept 1 of 2: Rank from the top and from the bottom

A person who is 3rd from the top has 2 people above. Counting from both ends counts that person twice, so take one away to get the total.

Definition

Total in the line $=$ rank from the top $+$ rank from the bottom $- 1$ .
Rank from the other end $=$ total $-$ rank from this end $+ 1$ .
People above a person ranked $r$ from the top: $r - 1$ .
People between two ranks from the same end: the larger rank minus the smaller, minus 1.

Total and rank from the other end

N = T + B - 1 \qquad B = N - T + 1

Nnumber of people in the line
Trank from the top
Brank from the bottom

Worked example

Asha is 12th from the top and 20th from the bottom of her class. How many students are in the class?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Order, Ranking and Arrangement · Ranks and Order Puzzles

Vishnu ranks 16th from the top and 49th from the bottom in a class. Then how many students are there in that class?

Adding the ranks without taking one away

Ranks from the top and from the bottom both count the same person, so their plain sum is one too many. The total is the sum minus 1.

Counting the end people as between

Between the 20th and the 24th there are only the 21st, 22nd and 23rd: three people, not four. Leave both named people out.

Concept 2 of 2: Build the line, then fill the grid

An order puzzle hands you pieces: A earns more than B, C less than D. Join the pieces into one line from top to bottom, then fit the jobs using what each clue rules out. Every question in the set is then read off the same line.

Definition

The method:

Turn each comparison into a short chain: 'more than' puts a person above, 'less than' below.
Join chains that share a person into one line.
More than only X means exactly one person is below, and it is X: X is at the bottom.
Earns the most puts that person at the top of everyone.
For jobs, draw a grid of people and jobs. Fill what the clues give, cross out what they forbid, and the last box in a row or column is forced.
Count the people named against the number the puzzle says. If they do not match, answer only what every reading agrees on.

Worked example

Five friends: A is taller than B. C is shorter than B. D is the tallest. E is taller than A but shorter than D. Who is in the middle?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q60Moderate

Example 2 · Order, Ranking and Arrangement · Ranks and Order Puzzles

Based on the information given, answer the following four questions. P, Q, R, S, T, U, V and W are the seven members of a family. There are three female members. Each of them has a different profession- Lawyer, Chartered Accountant (CA), Engineer, Teacher, Doctor, Architect and Pharmacist. No lady is either Pharmacist or CA. Each of them has a different monthly income. The CA earns the most. S, the Engineer earns less than V, the doctor. R, the teacher earns more than P and less than S. W's wife earns the least. T is unmarried lady lawyer and she earns less than P and more than only Q. The Pharmacist's income is not the lowest.

____ earns the least.

Reading 'more than only' as 'more than'

'More than only X' fixes X as the very lowest, with nobody else below. Plain 'more than X' leaves others possibly below too.

Answering from half the clues

A person you place from one clue can be moved by another. Join every comparison into one line before you answer any question in the set.

Trusting a puzzle that does not add up

If a puzzle names more people than it says there are, one person is left without a job or a place. Answer from what is fixed whichever person is the extra, and expect a key that rests on a guess.

Drill 3 more on build the line, then fill the grid

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Rank from the top and from the bottom
Total and rank from the other end
$N = T + B - 1 \qquad B = N - T + 1$

Watch out for (5)

Adding the ranks without taking one away
→ Rank from the top and from the bottom
Counting the end people as between
→ Rank from the top and from the bottom
Reading 'more than only' as 'more than'
→ Build the line, then fill the grid
Answering from half the clues
→ Build the line, then fill the grid
Trusting a puzzle that does not add up
→ Build the line, then fill the grid

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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