IMAT Logical Reasoning and Problem Solving · Formula sheet

Spatial and Sequence Reasoning formulas

11 formulas, 2 reference tables and 17 common traps for IMAT Logical Reasoning and Problem Solving Spatial and Sequence Reasoning, grouped by subtopic.

Full notes with worked examples

Number Sequences: Finding the Rule

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Arithmetic and geometric sequences

The nth term

a_n = a_1 + (n - 1)\,d \qquad a_n = a_1\, r^{\,n - 1}

$a_1$ first term
$d$ common difference (arithmetic)
$r$ common ratio (geometric)
$n$ position of the term you want

Recurrence rules: each term built from the terms before it

Two-term recurrence

a_n = p\,a_{n-1} + q\,a_{n-2}

$p, q$ fixed whole numbers, often 1 or 2
$a_{n-1}, a_{n-2}$ the previous term and the one before it

Common traps

The nth term is n minus 1 steps away

The first term is already in place, so the 7th term is only 6 steps on. Using

r^n

n \times d

gives the next term after the one asked for, and that value is usually among the options.

Check the rule on every given term

A rule that fits the first three terms can fail at the fourth. Before choosing an option, run your rule through all the terms shown; the wrong options are built from rules that fit only the beginning.

Use the term after the gap

When a missing term sits in the middle, the terms after it are a free check. A value that fits the rule on one side but not the other is the wrong rule, and IMAT distractors are often exactly those values.

Letter Sequences and Letter Codes

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Letter codes: shifts and reversed alphabets

Shift code

\text{coded position} = \text{position} + k \quad (\text{subtract } 26 \text{ if above } 26)

$k$ the shift; negative means move backwards

Alphabet positions and letter sequences

Letters	Positions	Positions from the end
A B C D E	1 2 3 4 5	26 25 24 23 22
F G H I J	6 7 8 9 10	21 20 19 18 17
K L M N O	11 12 13 14 15	16 15 14 13 12
P Q R S T	16 17 18 19 20	11 10 9 8 7
U V W X Y	21 22 23 24 25	6 5 4 3 2
Z	26	1

Learn the anchors E, J, O, T, Y and count from the nearest one.

Common traps

Position from the end is 27 minus the position

Z is 1st from the end, not 0th. So a letter at position

p

27 - p

from the end: M (13) is 14th from the end. Using 26 minus the position puts you one letter off, and that letter is usually an option.

Shifts wrap from Z back to A

With a shift of +3, X becomes A, not a symbol beyond Z. Forgetting to subtract 26 gives a position like 27, and an option that skips the wrap is often there to catch it.

Clock Hands, Elapsed Time and Days of the Week

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Clock hands: rotations, degrees and elapsed time

Converting hand movement to time

\text{minutes} = 60 \times (\text{minute-hand turns}) = 2 \times (\text{hour-hand degrees})

$6°$ minute hand, per minute
$0.5°$ hour hand, per minute

The angle between the hour hand and the minute hand

Angle between the hands

\theta = \lvert 30H - 5.5M \rvert \qquad (\text{if } \theta > 180°, \text{ use } 360° - \theta)

$H$ hour shown, 0 to 11
$M$ minutes past the hour

Days of the week and times across midnight

Weekday after N days

\text{shift} = N \bmod 7

$N \bmod 7$ the remainder when $N$ is divided by 7

Common traps

Degrees of the hour hand are not minutes

The hour hand moves only 0.5° per minute, so 75° of the hour hand is 150 minutes, not 75. Always say which hand has moved before converting.

The hour hand moves between the hours

At 7:30 the hour hand is not on the 7: it has moved

30 \times 0.5° = 15°

towards the 8. Leaving it on the hour mark gives an angle that is off by

0.5M

degrees.

Day 1 is the starting day

If a course starts on day 1, day 100 is 99 days later, not 100. Counting both ends (or neither) moves the answer one weekday, and that neighbouring day is always an option.

Mirror Images, Half Turns and Folded Paper

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Folding, punching and unfolding paper

Holes after folding and punching

\text{holes} = \text{punches} \times 2^{f}

$f$ number of folds in half (punches not on a fold line)

Letters and digits under mirrors and half turns

Change	Order of characters	Capital letters that look the same	Seven-segment digits
Mirror on a vertical line	Reversed	A H I M O T U V W X Y	0, 1 and 8 stay; 2 and 5 swap; the others are not digits
Mirror on a horizontal line	Unchanged	B C D E H I K O X	0, 1, 3 and 8 stay; 2 and 5 swap; the others are not digits
Half turn (180°)	Reversed	H I N O S X Z	0, 1, 2, 5 and 8 stay; 6 and 9 swap; 3, 4 and 7 are not digits

Letters assume plain capitals with no serifs. A seven-segment 1 moves to the other side of its space but still reads as 1.

Common traps

A mirror reverses the order, not only the symbols

Mirroring each digit in place, without reading the line from the other end, gives a wrong answer that looks convincing. A vertical mirror and a half turn both reverse the order; only a horizontal mirror keeps it.

A cut on the fold line makes one hole

When the cut touches the fold, the two halves of the hole are joined along that line and open into one shape, twice as wide. Counting it as two separate holes is wrong.

Painted Cubes, Cube Nets and Views of Solids

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Painted cubes: how many small cubes have 0, 1, 2 or 3 painted faces

Painted n × n × n cube

8 \;+\; 12(n-2) \;+\; 6(n-2)^2 \;+\; (n-2)^3 \;=\; n^3

$8$ corners, 3 faces painted
$12(n-2)$ edge cubes, 2 faces painted
$6(n-2)^2$ face cubes, 1 face painted
$(n-2)^3$ core, no paint

Common traps

"At least one face" includes the corners and edges

The quickest route is total minus the hidden core:

n^3 - (n-2)^3

. Answering with only the one-face cubes, or with the core itself, gives two of the usual wrong options.

Neighbours in the net are never opposite

Two squares that share an edge in the net always end up as neighbouring faces of the cube. The opposite face is two steps away in a straight line, or is found by elimination once the other two pairs are known.

A view shows the tallest stack, not the total

Cubes hidden behind a taller stack add nothing to a front view. Adding up a column gives the number of cubes in it, not the height you see, and that sum is a common wrong option.

Directions, Grid Paths, Seating Plans and Lengths

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Compass directions, turns and distance from the start

Distance from the start

d = \sqrt{x^2 + y^2}

$x$ net distance east (negative for west)
$y$ net distance north (negative for south)

Grids: counting shortest routes and reading numbered boards

Shortest grid routes

\text{routes} = \binom{m + n}{n} = \frac{(m + n)!}{m!\, n!}

$m$ blocks east (or right)
$n$ blocks north (or up)

Lengths along a line: fence posts and overlapping pieces

Overlapping pieces in a row

\text{length} = n\,\ell - (n - 1)\,o

$n$ number of pieces
$\ell$ length of one piece
$o$ overlap at each join

Common traps

Right and left depend on the way you face

A right turn while facing south sends you west, not east. Update the facing direction after every turn before reading the next instruction; the wrong options usually come from treating right as always east.

Count orders of moves, not moves

Every shortest route has the same number of moves, so the number of moves is never the number of routes. The routes differ only in the order of east and north steps, which is what

\binom{m+n}{n}

counts.

Facing the centre flips left and right on the far side

A person across the table from you has their left on your right. Seen from above with everyone facing in, "to the left of" always means the next seat clockwise; using your own left and right for everyone gives the mirror-image arrangement.

Posts are one more than gaps on a line, equal on a loop

Dividing the length by the spacing counts gaps. On an open line add one post for the far end; on a closed loop do not. Each version gives a wrong answer one away from the right one.

Spatial and Sequence Reasoning formulas

Number Sequences: Finding the Rule

Arithmetic and geometric sequences

Recurrence rules: each term built from the terms before it

Common traps

Letter Sequences and Letter Codes

Letter codes: shifts and reversed alphabets

Alphabet positions and letter sequences

Common traps

Clock Hands, Elapsed Time and Days of the Week

Clock hands: rotations, degrees and elapsed time

The angle between the hour hand and the minute hand

Days of the week and times across midnight

Common traps

Mirror Images, Half Turns and Folded Paper

Folding, punching and unfolding paper

Letters and digits under mirrors and half turns

Common traps

Painted Cubes, Cube Nets and Views of Solids

Painted cubes: how many small cubes have 0, 1, 2 or 3 painted faces

Common traps

Directions, Grid Paths, Seating Plans and Lengths

Compass directions, turns and distance from the start

Grids: counting shortest routes and reading numbered boards

Lengths along a line: fence posts and overlapping pieces

Common traps

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