Embedded Figures & Figure Counting

1X embeds only in the same orientation

An embedded-figure question on SSC CGL Tier-1 is worded almost word for word: 'Select the option figure in which the given figure (X) is embedded (rotation is NOT allowed).' Figure (X) is a small line drawing printed beside four option drawings. Hold one concrete X for this topic: an L-hook — a vertical stroke two units tall with a one-unit foot sticking out to the right, the same shape as a capital L. Embedded means every stroke of that L-hook appears as a subset of the option's strokes, in the same arrangement. Extra strokes in the option are allowed; a missing or shifted stroke of X is not.

The parenthetical is the trap. Rotation is not allowed, so X must sit in the option the way it was printed — long arm vertical, foot pointing right. An option that contains a perfect copy of the L-hook turned 90 degrees clockwise (a long arm running horizontally, a short tick hanging down) looks right to anyone who rotated X in their head. A rotated copy of X is a trap, even when every edge of X is present. The test is not 'could this be X after a turn'; it is 'is this X, as printed'.

This is a different skill from the figural series, analogy and odd-figure work on Non-Verbal Reasoning. Those questions ask you to name a transformation and apply it. An embedded-figure question forbids the transformation: you are hunting a copy, not inventing a turn. If an option looks like X only after you tilt the page, leave it.

Slide, do not turn

1. Name X's armsSay the orientation out loud: for the L-hook, long arm vertical, foot to the right. That sentence is the rule the stem locked.
2. Slide a tracingMove a mental copy of X across the option without rotating it. A fit is a match; a fit that needs a tilt is not.
3. Reject rotated copiesAn option whose hook is the same L turned 90 degrees is built for the reader who ignored 'rotation is NOT allowed'. Strike it even if every edge of X is there.
4. Ignore extra strokesCamouflage lines around a correctly oriented X do not disqualify it. Missing or moved strokes of X do.

2Scan the edges of X, not the whole option

The same L-hook is still X; the option is now a cluttered line drawing — extra boxes, diagonals, stubs — with X sitting somewhere inside as a subset of those strokes. Staring at the whole option is how you miss it. The option is larger than X on purpose, and most of its ink is camouflage.

The distinctive piece of this X is the corner where the two-unit vertical meets the one-unit right foot. Find that corner in the same orientation, then walk both arms: the long edge still vertical, the foot still pointing right. If both arms are present, X is embedded, no matter how many extra strokes crowd the rest of the card. If either arm is missing or the corner has been turned, it is not X.

So scan the edges of X, not the whole option. Matching a silhouette is slow and it is what the rotated-hook distractor is counting on. Matching two named edges in the printed orientation is the method that survives clutter.

Corner, then arms

1. Name the cornerFor the L-hook, the load-bearing joint is where the two-unit vertical meets the one-unit right foot. That is the first thing you hunt.
2. Walk both armsFrom that corner, confirm the long edge is still vertical and the foot still points right — the same orientation the stem printed.
3. Treat extras as camouflageA closed box in the opposite corner, a diagonal, a V — none of those disqualify a correctly oriented X. They are there to slow a whole-option stare.

3Count unit pieces, then composites, then the whole

The CGL stem is 'How many triangles are there in the given figure?' — or squares, in the same shape of question. The figure is a line drawing, and the answer is never just the smallest pieces you can see at a glance. Every larger triangle (or square) made of those pieces also counts. Skipping the composites is how the unit-only trap is written into the options.

Hold one concrete figure: triangle ABC with two rays from A down to points P and Q that cut BC into three equal parts. The unit triangles are the three with one base gap: ABP, APQ, AQC. The two-unit triangles are the two that span two adjacent gaps: ABQ and APC. The large one is ABC itself. Count the unit pieces first, then each larger composite, then the whole figure: 3 + 2 + 1 = 6. The same order works on a square with both diagonals drawn: four unit triangles around the centre, then four two-unit half-squares, total 8 — there is no larger triangle wrapping the whole square.

The closed form for this split-base family is n(n+1)/2 when n-1 rays cut the base into n parts. For n = 3 that is 6, matching the enumeration. Use the formula after the size classes are listed, not instead of them — a formula you cannot point at in the figure is a guess.

Size classes, then total

1. List the unit piecesName every smallest triangle (or square) that uses one cell of the figure. In the split-base triangle that is ABP, APQ, AQC — 3.
2. List each larger sizeTwo adjacent units make a 2-unit triangle (ABQ, APC — 2). Keep going until the size that is the whole figure (ABC — 1).
3. Add the classes3 + 2 + 1 = 6. Check against n(n+1)/2 with n = 3 only after the list exists.
4. Spot the unit-only optionThe options will include 3. That is the unit count, not the answer.

4Squares in an n-by-n grid

The same CGL stem — 'How many squares are there in the given figure?' — on a grid is a formula once you have seen the size classes. An n \times n grid means n unit squares along each side, so (n+1) grid lines each way. Size-1 squares are the unit cells: n \times n of them. Size-2 squares are 2 \times 2 blocks of unit cells; their top-left corner can sit in (n-1) rows and (n-1) columns, so (n-1)^2 of them. Size-k in general has (n-k+1)^2 positions. Add the classes: 1^2 + 2^2 + \cdots + n^2, which is also n(n+1)(2n+1)/6.

Work the 3-by-3 case that the figure shows. Size 1: 3 \times 3 = 9. Size 2: 2 \times 2 = 4 positions. Size 3: one large square. Total 9 + 4 + 1 = 14, and 1^2 + 2^2 + 3^2 = 14 agrees. Counting only the unit squares is the trap: a 3-by-3 grid holds fourteen squares, not nine. Nine is sitting in the options because it is the number you get if you stop after the unit cells.

The formula does not replace the size-class list on a non-grid figure — a square with extra diagonals is the previous concept's enumeration, not this sum of squares. Reach for 1^2 + \cdots + n^2 only when the drawing is a plain unit grid.

Figure. A 3-by-3 grid of unit squares. Size 1 contributes 9, size 2 contributes 4, size 3 contributes 1. Total 14, not 9.

Classes, then the sum

1. Read nCount unit cells along one side. A 3-by-3 grid has n = 3, not 4 — 4 is the number of grid lines.
2. Count each size kSize k has (n-k+1)^2 positions. For n = 3: size 1 gives 9, size 2 gives 4, size 3 gives 1.
3. Sum, then check the formula9 + 4 + 1 = 14, and 1^2 + 2^2 + 3^2 = 14. The closed form n(n+1)(2n+1)/6 with n = 3 is the same 14.
4. Reject the unit-only optionNine is the size-1 count. The stem asked for squares, which includes the four 2-by-2 blocks and the large one.

5Name the one changing attribute, then pick the missing cell

A missing-figure or pattern-completion item on CGL shows a strip or a 2-by-2 (sometimes 3-by-3) of small frames with one cell blank, and asks you to select the figure that completes the pattern. It looks like a figural series, and it is not the same job as Non-Verbal Reasoning's multi-stream series. There the next frame applies several tracked changes at once. Here the intended rule is usually one changing attribute — a count, a position, or a shading toggle — and the distractors apply a different attribute (most often a rotation) to a frame that otherwise looks busy enough to be right.

The method is to name the one changing attribute before you touch the options. Say it as a sentence with a number in it: 'the inner tick count rises by one each cell, orientation fixed.' Then the blank cell is that sentence applied once more. An option that instead rotates the outer square 90 degrees and keeps the old tick count has answered a rotation question the stem did not ask. If two attributes both seem to move, you are on the Non-Verbal Reasoning series method; come back and check whether one of them is decoration.

This concept does not draw the four-frame exam strip. Approximating those frames with boxes would invent the item. The attack order lives in the steps and in the 'Attributes that complete a pattern' table: name one attribute, predict the missing cell, reject any option that applies a different attribute.

The exam picture is a row or 2-by-2 of small frames with one cell blank — for example three cells showing 1, then 2, then 3 inner ticks, and a question-mark cell. No drawing accompanies this concept. Read the 'Attributes that complete a pattern' table as the figure: one row per attribute, the missing cell continuing that row, and the distractor applying a different row (usually rotation).

Name, predict, reject

1. Name one attributeCount, position or shading — pick the one that actually changes across the given cells, and say it in a sentence with a number ('inner ticks: 1, then 2, then 3').
2. Predict the blankApply that sentence once more. If the rule was +1 tick with orientation fixed, the missing cell has 4 ticks and the same orientation.
3. Reject a different attributeAn option that rotates or reflects the frame, or that keeps the old count, has answered a different question. Strike it even if it 'looks like the set'.