IMAT Logical Reasoning · Spatial and Sequence Reasoning

Directions, Grid Paths, Seating Plans and Lengths

Track position and facing direction step by step, count grid routes by choosing moves, fix one person at a table before placing the rest, and count gaps rather than posts.

All Spatial and Sequence Reasoning notes

Why this matters

A 2025 ministry question placed four people around a table using left and right and asked what can be deduced. The Cambridge papers numbered squares on a board (2014) and houses on a street (2012), asked how few sensors cover a grid (2022), and measured lengths along a chain and around a box (2011, 2019).

Concept 1 of 4: Compass directions, turns and distance from the start

Keep two things as you go: where you are and which way you face. A right turn moves your facing one step clockwise round the compass (north, east, south, west), a left turn one step back. At the end, the net distance east and north gives the straight-line distance by Pythagoras.

Definition

Clockwise order: north, east, south, west. A right turn is 90° clockwise; a left turn is 90° anticlockwise; an about-turn is 180°.
Record each walk as a change in east ( $x$ ) and north ( $y$ ); a walk west or south is negative.
The straight-line distance from the start is $\sqrt{x^2 + y^2}$ .
Common right-angled triangles: 3, 4, 5 and 6, 8, 10 and 5, 12, 13.

Distance from the start

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

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

Worked example

Starting out facing north, a hiker walks 4 km, turns right and walks 3 km, then turns right again and walks 8 km. How far is she from the start, and in which direction from it?

Practice this conceptself-check · 3 quick reps

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.

Concept 2 of 4: Grids: counting shortest routes and reading numbered boards

A shortest route across a street grid is just a list of moves, some east and some north. Every route uses the same number of each; routes differ only in the order. So counting routes means counting the ways to choose which moves are the north ones.

Definition

A shortest route that goes $m$ blocks east and $n$ blocks north has $m + n$ moves, and the number of such routes is $\binom{m+n}{n}$ .
$\binom{a}{b} = \dfrac{a!}{b!\,(a-b)!}$ ; for example $\binom{5}{2} = 10$ .
Numbered boards: find the row of a number by dividing by the row length. On a "snake" board the direction of counting reverses on every row.
Covering a grid: if each piece covers $k$ squares, you need at least the number of squares divided by $k$ , rounded up.

Shortest grid routes

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

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

Worked example

How many shortest routes are there from one corner of a 2 by 2 block grid to the opposite corner?

Practice this conceptself-check · 3 quick reps

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.

Concept 3 of 4: Seating around a table: left and right when facing the centre

People round a table face the middle, so their left and right are not yours. Seen from above, a person's left-hand neighbour is the next seat clockwise, and their right-hand neighbour is the next seat anticlockwise. Fix one person first; with a round table, rotating everyone changes nothing.

Definition

For people seated round a table, all facing the centre, seen from above:

The person immediately to someone's left is the next seat clockwise.
The person immediately to someone's right is the next seat anticlockwise.
With an even number of evenly spaced seats, the person opposite is half-way round.
Number the seats clockwise from one fixed person, then place the others from the most definite clue.
For people in a row facing you, their left is your right.

Worked example

Lia, Max, Nora and Omar sit at a round table with four evenly spaced seats, all facing the centre. Lia sits opposite Max, and Nora sits immediately to Lia's right. Who sits immediately to Omar's left?

Practice this conceptself-check · 3 quick reps

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.

Concept 4 of 4: Lengths along a line: fence posts and overlapping pieces

Count gaps, not objects. Along a straight fence the posts outnumber the gaps by one, because both ends need a post. Around a closed loop the last gap returns to the first post, so posts and gaps are equal. When pieces overlap, each join loses one overlap from the total.

Definition

A straight line of length $L$ with a post every $g$ , including both ends: $L/g + 1$ posts.
A closed loop of length $L$ with a post every $g$ : $L/g$ posts.
Cutting a pole into $k$ pieces takes $k - 1$ cuts.
$n$ pieces of length $\ell$ laid in a row, each overlapping the next by $o$ , cover $n\ell - (n - 1)\,o$ : there are $n - 1$ joins.

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

Worked example

A straight fence 45 m long has a post every 2.5 m, including one at each end. How many posts are there? How many flags are needed round a 400 m running track with a flag every 25 m?

Practice this conceptself-check · 3 quick reps

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.

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 (3)

Compass directions, turns and distance from the start
Distance from the start
$d = \sqrt{x^2 + y^2}$
Grids: counting shortest routes and reading numbered boards
Shortest grid routes
$\text{routes} = \binom{m + n}{n} = \frac{(m + n)!}{m!\, n!}$
Lengths along a line: fence posts and overlapping pieces
Overlapping pieces in a row
$\text{length} = n\,\ell - (n - 1)\,o$

Watch out for (4)

Right and left depend on the way you face
→ Compass directions, turns and distance from the start
Count orders of moves, not moves
→ Grids: counting shortest routes and reading numbered boards
Facing the centre flips left and right on the far side
→ Seating around a table: left and right when facing the centre
Posts are one more than gaps on a line, equal on a loop
→ Lengths along a line: fence posts and overlapping pieces

Drill the past-year questions on this topic

Open the bank with this topic pre-filtered.

Open in the bank