Reasoning Formulas and Counting Shortcuts: Ranking, Handshakes, Triangles, Squares, Clock and Opposite Letters for SSC
Rahul Kumar
SSC Exam Expert & Content Editor
In this note
- 1.Ranking and Position
- 2.Counting Formulas
- 3.Clock and Calendar
- 4.Letters, Numbers and Coding
- 5.Dice, Cubes and Directions
- 6.Venn Diagrams and Logic
- 7.Solved Examples
- 8.Quick Tips
- 9.Quick Revision Points
- 10.Practice Questions With Answers
- 11.How to Practise This Chapter
- 12.Frequently Asked Questions
PDF Notes: Reasoning Formulas and Counting Shortcuts: Ranking, Handshakes, Triangles, Squares, Clock and Opposite Letters for SSC
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Reasoning looks like a subject without formulas, but many question types have short formulas that save a lot of time. This reasoning formulas and counting shortcuts guide for SSC collects them in tables: ranking, handshakes, counting of figures, the chess board, the clock and calendar, letters and dice, with solved examples, practice questions and FAQs.
Ranking and Position
| Case | Formula |
|---|---|
| Total persons in a row | Rank from the top + rank from the bottom − 1 |
| Rank from the bottom | Total − rank from the top + 1 |
| Number of persons between A and B | Difference of their ranks − 1 |
| Person X is m-th from the left and n-th from the right | Total = m + n − 1 |
| After the exchange of positions | Use the difference of the new and the old positions to find the number of persons between |
Counting Formulas
| Item | Formula |
|---|---|
| Handshakes among n persons (each pair once) | n(n − 1)/2 |
| Matches in a league among n teams (each pair once) | n(n − 1)/2; in a double league n(n − 1) |
| Knock-out matches for n teams | n − 1 |
| Number of diagonals of an n-sided polygon | n(n − 3)/2 |
| Triangles in a figure formed by n lines drawn from a vertex to the base (n lines inside) | (n + 1)(n + 2)/2 |
| Squares in an n × n grid | 1² + 2² + … + n² = n(n + 1)(2n + 1)/6 |
| Rectangles in an m × n grid (including squares) | m+1C2 × n+1C2 |
| Rectangles in an n × n grid | [n(n + 1)/2]² |
| Squares on a chess board (8 × 8) | 204 |
| Rectangles on a chess board (8 × 8, including squares) | 1,296 |
| Maximum number of regions formed by n lines in a plane | n(n + 1)/2 + 1 |
| Maximum points of intersection of n lines | n(n − 1)/2 |
Clock and Calendar
| Item | Formula |
|---|---|
| Angle between the hands at H hours M minutes | |30H − 5.5M| |
| Hands overlap after H o’clock at | 60H/11 minutes past H |
| Hands overlap in a day | 22 times |
| Hands at right angles in a day | 44 times |
| Hands opposite in a day | 22 times |
| Odd days | Ordinary year 1; leap year 2; 100 years 5; 200 years 3; 300 years 1; 400 years 0 |
| Leap year | Divisible by 4; a century is a leap year only if divisible by 400 |
| Day after n days | Divide n by 7 and move the day by the remainder |
Letters, Numbers and Coding
| Item | Rule |
|---|---|
| Position of the letter A to Z | 1 to 26; E = 5, J = 10, O = 15, T = 20, Z = 26 |
| Opposite letter | 27 − position (A↔Z, B↔Y, C↔X…) |
| Sum of the positions of EJOTY | 5, 10, 15, 20 and 25; useful marks in the alphabet |
| Number of letter pairs in a word with the same gap as in the alphabet | Count the pairs where the position difference equals the word distance |
| Sum of the first n natural numbers | n(n + 1)/2 |
| Squares and cubes | Learn up to 30 and 15 |
Dice, Cubes and Directions
| Item | Rule |
|---|---|
| Opposite faces of a standard dice | Add up to 7: 1–6, 2–5, 3–4 |
| A cube | 6 faces, 12 edges, 8 vertices |
| Painted cube of side n cut into unit cubes | 3 faces: 8; 2 faces: 12(n − 2); 1 face: 6(n − 2)²; 0 faces: (n − 2)³ |
| Cuts to make n³ cubes from a cube | 3(n − 1) |
| Distance in direction sense | Use the Pythagoras theorem for the right-angled paths (3, 4, 5; 5, 12, 13; 8, 15, 17) |
| Turns in directions | A left turn from North gives West; a right turn gives East; the opposite of North is South |
Venn Diagrams and Logic
| Item | Rule |
|---|---|
| Two sets | n(A ∪ B) = n(A) + n(B) − n(A ∩ B) |
| Three sets | n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C) |
| Syllogism contradictories | All and Some-not; No and Some |
| Conditional statement | If P then Q is equivalent to If not Q then not P |
Solved Examples
Example 1. A is 7th from the top and 18th from the bottom in a class. Find the total number of students.
7 + 18 − 1 = 24.
Example 2. How many handshakes are there among 10 persons?
10 × 9/2 = 45.
Example 3. How many squares are there on a chess board?
1² + 2² + … + 8² = 8 × 9 × 17/6 = 204.
Example 4. How many triangles are there in a triangle in which 3 lines are drawn from a vertex to the base?
(3 + 1)(3 + 2)/2 = 10.
Example 5. Find the angle between the hands of a clock at 4:20.
|30 × 4 − 5.5 × 20| = |120 − 110| = 10°.
Example 6. What is the opposite letter of K?
27 − 11 = 16, which is P.
Quick Tips
- Memorise the ranking formula and the handshake formula.
- Learn the counting formulas of the grids and the chess board.
- Use the opposite letter rule 27 minus the position.
- Use the clock formula and not the drawing.
Quick Revision Points
- Total = top rank + bottom rank − 1.
- Handshakes = n(n − 1)/2.
- A chess board has 204 squares.
- Clock angle = |30H − 5.5M|.
- Opposite letter = 27 − position.
Practice Questions With Answers
- A student is 12th from the top and 20th from the bottom. How many students are there?
Answer: 31 - How many handshakes among 15 persons?
Answer: 105 - How many matches in a knock-out tournament of 16 teams?
Answer: 15 - How many squares in a 5 × 5 grid?
Answer: 55 - How many rectangles in a 3 × 3 grid?
Answer: 36 - Angle between the hands at 6:00?
Answer: 180° - Opposite letter of H?
Answer: S - How many diagonals does a decagon have?
Answer: 35 - In a standard dice, what is opposite to 2?
Answer: 5 - Triangles in a triangle with 4 lines from a vertex to the base?
Answer: 15
How to Practise This Chapter
Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Formula Guide notes page, read the previous topic Fractions, Percentages and Tables Quick Reference.
Frequently Asked Questions
What is the ranking formula?
Total = rank from top + rank from bottom − 1.
How many handshakes are there among n persons?
n(n − 1)/2.
How many squares are there on a chess board?
204.
How many rectangles are there on a chess board?
1,296.
What is the formula for the angle between the clock hands?
|30H − 5.5M|.
What is the opposite letter rule?
The opposite of the letter at position n is at position 27 − n.
What do the opposite faces of a dice add up to?
7.
How many knock-out matches are needed for n teams?
n − 1.
How do I count the squares in an n by n grid?
Add the squares of 1 to n, n(n + 1)(2n + 1)/6.
Is this topic asked in SSC?
Yes. These formulas are used in the SSC reasoning questions.
How many times do the clock hands overlap in a day?
22 times.
How many diagonals does a polygon of n sides have?
n(n − 3)/2.
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.