General Mental Ability — Quick Revision

Every rule and formula from Series and Analogies, Puzzles, Calendars and Clocks on one page, for the last week before Prelims Paper II.

Series and Analogies

The five-step check for number series

Step	Test	Example → next
1	Differences (then second differences)	3, 7, 13, 21, 31 → 43
2	Ratios / \( \times k \pm c \)	5, 11, 23, 47, 95 → 191
3	Squares, cubes and neighbours	2, 6, 12, 20, 30 → 42
4	Primes	2, 3, 5, 7, 11, 13 → 17
5	Alternating (two series)	2, 20, 4, 17, 6, 14, 8 → 11

Standard sequences

Rule	Terms
\( n^2 \)	1, 4, 9, 16, 25, 36, 49, 64, 81, 100
\( n^3 \)	1, 8, 27, 64, 125, 216, 343
\( n(n+1) \)	2, 6, 12, 20, 30, 42, 56
\( \frac{n(n+1)}{2} \) (triangular)	1, 3, 6, 10, 15, 21, 28
Factorials	1, 2, 6, 24, 120, 720
Primes below 50	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Letters

$$\boxed{\text{E J O T Y} = 5,\ 10,\ 15,\ 20,\ 25} \qquad \boxed{\text{Opposite letter} = 27 - \text{position}}$$

Opposite pairs: A–Z, B–Y, C–X, D–W, E–V, F–U, G–T, H–S, I–R, J–Q, K–P, L–O, M–N.

Analogy method: say the relation as a sentence (“Hirakud is a dam on the Mahanadi”), then test each option with the same sentence.

Puzzles

$$\boxed{\text{Total in a row} = \text{Left} + \text{Right} - 1} \qquad \boxed{\text{Rank from other end} = \text{Total} - \text{Rank} + 1}$$

Puzzle	Formula
People between A and B (no overlap)	Total \( - (L_A + R_B) \)
Interchange	New position of one = old position of the other
Handshakes / round-robin league	\( \frac{n(n-1)}{2} \)
Knock-out tournament	\( n - 1 \) matches
Heads and legs (hens, cows)	Cows \( = \frac{\text{Legs} - 2\times\text{Heads}}{2} \)

Painted cube cut into \( n^3 \) pieces

Faces painted	3	2	1	0
Formula	8	\( 12(n-2) \)	\( 6(n-2)^2 \)	\( (n-2)^3 \)
\( n = 4 \)	8	24	24	8
\( n = 5 \)	8	36	54	27

Grid puzzles: draw the table → direct clues → negative clues → linking clues → check every clue.

Calendars

$$\boxed{\text{Leap year: divisible by 4; century year only if divisible by 400}}$$

Period	Odd days	Period	Odd days
Ordinary year	1	100 years	5
Leap year	2	200 years	3
31-day month	3	300 years	1
30-day month	2	400 years	0
February	0 (1 in leap year)

Odd days	0	1	2	3	4	5	6
Day	Sun	Mon	Tue	Wed	Thu	Fri	Sat

$$\boxed{\text{Day of any date: complete years} + \text{complete months} + \text{days, then} \div 7}$$

Fact	Value
Calendar repeat	Leap year 28 years; year after a leap year 6; the other two 11
Same-calendar months (ordinary year)	Jan–Oct; Feb–Mar–Nov; Apr–Jul; Sep–Dec
Same-calendar months (leap year)	Jan–Apr–Jul; Feb–Aug; Mar–Nov; Sep–Dec
Doomsday 2026	Saturday (4/4, 6/6, 8/8, 10/10, 12/12, 9/5, 5/9, 7/11, 11/7, 14 March, last day of February)
15 Aug 1947 / 26 Jan 1950 / 1 Nov 2000	Friday / Thursday / Wednesday

Clocks

$$\boxed{\theta = \left|\,30H - \frac{11}{2}M\,\right| \quad (\text{if } \theta > 180^\circ,\ \text{use } 360^\circ - \theta)}$$$$\boxed{k \text{ minute-spaces apart between } H \text{ and } H+1:\ M = (5H \pm k)\times\frac{12}{11}}$$

Fact	Value
Hand speeds	Minute 6°/min, hour 0.5°/min, relative 5.5°/min
\( k \) for coincide / right angle / opposite	0 / 15 / 30
Hands coincide every	\( 65\frac{5}{11} \) minutes
In 24 hours	22 coincide, 22 opposite, 44 right angles, 44 straight lines
Mirror image	\( 11{:}60 - \) time (\( 23{:}60 \) for 12:xx)
Faulty clock correct again	\( \frac{720}{\text{gain per hour}} \) hours
Hands meet more often than every \( 65\frac{5}{11} \) min	Clock is gaining

Don’t Confuse

This	vs	That
\( n(n+1) \): 2, 6, 12, 20		\( \frac{n(n+1)}{2} \): 1, 3, 6, 10
Round-robin \( \frac{n(n-1)}{2} \)		Knock-out \( n - 1 \)
100 years = 5 odd days		1 ordinary year = 1 odd day
Ordinary-year calendar repeats after 6 or 11 years		Leap-year calendar after 28 years
Hands coincide 22 times a day		Right angles 44 times a day
Mirror time \( 11{:}60 - t \)		Angle \( \lvert 30H - 5.5M \rvert \)