Visual Reasoning Aptitude

Cheatsheet Content

### Series Completion Visual series completion involves identifying the pattern in a sequence of figures and predicting the next figure. #### 1. Common Patterns - **Rotation:** Figures rotate clockwise (CW) or counter-clockwise (CCW) by a fixed angle (e.g., 45°, 90°, 180°). - **Reflection/Mirror Image:** Figures are reflected horizontally or vertically. - **Movement/Displacement:** Elements within a figure move to new positions (e.g., one step CW, two steps CCW). - **Addition/Removal of Elements:** Parts are added or removed in a sequence. - **Change in Size/Shape:** Elements grow, shrink, or transform into different shapes. - **Shading/Color Change:** Shaded parts move, change pattern, or alternate. - **Combination of Patterns:** Often, multiple patterns occur simultaneously. #### 2. Strategy 1. **Observe Changes:** Compare the first two figures, then the second and third, and so on. 2. **Identify Single Element Changes:** Focus on one element at a time (e.g., the arrow, the dot, the square). 3. **Note Direction & Magnitude:** How much and in what direction does it change? 4. **Confirm Pattern:** Ensure the identified pattern holds for all figures in the series. 5. **Predict Next Figure:** Apply the pattern to the last given figure to find the answer. #### 3. Example **Series:** Figure 1: Square with a dot at top-left. Figure 2: Square with a dot at top-right. Figure 3: Square with a dot at bottom-right. Figure 4: Square with a dot at bottom-left. **Pattern:** The dot moves 90° clockwise around the corners of the square. **Next Figure:** Square with a dot at top-left (repeats the cycle). ### Analogy Visual analogies require identifying the relationship between a pair of figures and applying that same relationship to another figure to find its missing pair. #### 1. Types of Relationships - **Rotation:** Figure B is a rotated version of Figure A. - **Reflection:** Figure B is a mirror image of Figure A. - **Size/Shape Change:** Figure B is a larger/smaller or transformed version of Figure A. - **Addition/Removal:** Elements are added or removed. - **Shading Change:** Shading pattern changes (e.g., inverted, moved). - **Superimposition:** Figure B is formed by combining elements of Figure A. - **Components:** Figure B shows a part of Figure A, or vice-versa. - **Orientation:** Changes in direction or alignment. #### 2. Strategy 1. **Analyze the First Pair (A:B):** Understand precisely how Figure B relates to Figure A. What transformations, additions, or deletions occurred? 2. **Identify the Rule:** Formulate a clear rule that transforms A into B. 3. **Apply Rule to Third Figure (C:?):** Apply the *exact same rule* to Figure C to find the missing Figure D. 4. **Check All Options:** Sometimes multiple options might seem plausible; choose the one that *best* fits the derived rule. #### 3. Example **A:B :: C:?** A: A large square. B: Two smaller squares inside the large square. C: A large circle. ?: **Relationship (A:B):** The large outer shape is replaced by two smaller versions of itself, placed inside the original boundary. **Applying to C:** The large circle should be replaced by two smaller circles inside the original circle's boundary. ### Classification (Odd One Out) In classification, you are given a set of figures, and you must identify the one that does not share a common characteristic or pattern with the others. #### 1. Common Distinguishing Features - **Number of Sides/Elements:** One figure might have a different count. - **Symmetry:** One figure might be asymmetrical while others are symmetrical. - **Rotation/Orientation:** One figure might be rotated differently compared to the pattern of others. - **Shading/Pattern:** One figure's shading might be inconsistent. - **Internal/External Relationship:** The relationship between internal and external elements might be different for one figure. - **Connectivity:** Elements might be connected differently. - **Position/Movement:** If elements are moving, one might break the sequence. #### 2. Strategy 1. **Examine All Figures:** Look at each figure individually and compare it to the others. 2. **Look for Commonalities:** Identify features that *most* figures share. 3. **Find the Discrepancy:** The figure that lacks this common feature or exhibits a unique one is the odd one out. 4. **Test Hypotheses:** If you suspect a pattern (e.g., "all figures have 4 sides"), quickly check if all figures except one fit this rule. #### 3. Example **Figures:** 1. A square. 2. A triangle. 3. A pentagon. 4. A circle. **Commonality:** Figures 1, 2, and 3 are polygons (closed shapes made of straight line segments). **Odd One Out:** Figure 4 (Circle) is not a polygon as it has no straight sides. ### Figure Formation / Grouping Identical Figures This type involves grouping a set of given figures into classes based on some common property, or identifying how smaller pieces can form a larger figure. #### 1. Types - **Grouping:** Categorizing figures based on shared properties (e.g., number of elements, type of lines, symmetry). - **Formation:** Identifying which set of smaller pieces can be assembled to form a given larger figure. This often involves mental rotation and rearrangement. #### 2. Strategy for Grouping 1. **Analyze Each Figure:** Understand its basic components, lines, curves, shading, and overall structure. 2. **Identify Potential Categories:** Look for characteristics that could serve as grouping criteria (e.g., "all figures with curved lines," "all figures with an even number of elements"). 3. **Sort Figures:** Assign each figure to its appropriate group. Ensure each group has a clear, consistent rule. #### 3. Strategy for Formation 1. **Examine the Target Figure:** Note its shape, specific angles, and any internal divisions. 2. **Examine the Component Pieces:** Understand the shapes and sizes of the pieces provided. 3. **Mental Manipulation:** Mentally rotate, flip, and move the component pieces to see if they fit together to form the target figure. 4. **Eliminate Options:** If a piece doesn't fit or leaves gaps/overlaps, eliminate that option. 5. **Focus on Edges/Corners:** Pay attention to how the edges and corners of the pieces align with the target figure. #### 4. Example (Formation) **Target Figure:** A large square divided diagonally into two triangles. **Option A Pieces:** Two identical right-angled triangles. **Option B Pieces:** A square and a rectangle. **Solution:** Option A pieces can be arranged to form the target figure. ### Embedded Figures (Hidden Figures) You are given a complex figure and a simple figure. You need to identify if the simple figure is hidden or embedded within the complex figure, often in a rotated or inverted form. #### 1. Strategy 1. **Study the Simple Figure:** Memorize its exact shape, angles, and proportions. Pay attention to unique features. 2. **Scan the Complex Figure Systematically:** Start from one corner and systematically scan the complex figure. 3. **Look for Key Features:** Instead of trying to find the whole figure at once, look for prominent parts of the simple figure (e.g., a specific angle, a unique curve, a parallel line pair). 4. **Allow for Rotation/Inversion:** Remember the simple figure might be rotated (CW/CCW) or inverted (mirror image) within the complex figure. Mentally rotate and flip the simple figure as you search. 5. **Trace with Your Finger/Pencil:** If allowed, tracing the outline of the simple figure within the complex one can be very helpful. #### 2. Common Pitfalls - **Distractors:** The complex figure often contains many lines and shapes designed to distract you. - **Focusing on Size:** The embedded figure might appear slightly larger or smaller due to the surrounding lines, but its *proportions* and *angles* must remain the same. - **Missing Rotations:** Forgetting to check for rotated or inverted versions. #### 3. Example **Simple Figure:** A capital 'T' shape. **Complex Figure:** A grid-like pattern with many intersecting vertical and horizontal lines, and some diagonal lines. **Solution:** Scan the grid for a vertical line with a horizontal line bisecting its top end. ### Paper Folding and Cutting This section tests your ability to visualize the outcome of folding a piece of paper, making cuts, and then unfolding it. #### 1. Strategy for Folding 1. **Visualize Each Fold:** Mentally perform each fold exactly as described. Imagine the paper getting smaller and layering up. 2. **Mark Reference Points:** If the paper has initial markings (like a dot or a corner cut), keep track of where these reference points move with each fold. 3. **Understand Layers:** After several folds, understand how many layers of paper are at each point where a cut will be made. #### 2. Strategy for Cutting 1. **Locate the Cut:** Identify where the cut is made on the *folded* paper. 2. **Determine Cut Shape:** Understand the shape of the cut. 3. **Trace the Cut on Layers:** Mentally trace the cut through all the layers of paper at that specific point. Each layer will have the cut. #### 3. Strategy for Unfolding 1. **Reverse the Folds:** Unfold the paper in the reverse order of how it was folded. 2. **Mirror Image Effect:** Each time you unfold, the cuts made will appear as a mirror image across the fold line. 3. **Symmetry:** Cuts made on a folded edge will often result in symmetrical patterns when unfolded. Cuts made in the center of the folded paper will appear in the center of the unfolded paper. #### 4. Example **Steps:** 1. A square sheet of paper is folded in half vertically (right side over left). 2. It is then folded in half horizontally (bottom half over top). 3. A small triangular cut is made at the bottom-right corner of the *folded* paper. **Unfolding:** - When unfolded horizontally, the triangular cut will create two triangles along the horizontal fold line. - When unfolded vertically, these two triangles will be mirrored, resulting in four triangles forming a diamond shape in the center of the original square. ### Dot Situation In dot situation problems, you are given a figure with one or more dots placed in specific regions (e.g., inside a circle, a square, and a triangle simultaneously). You then need to choose an option figure where the dot(s) can be placed in the *exact same* type of region(s). #### 1. Strategy 1. **Analyze the Original Figure:** For each dot, identify *all* the basic geometric shapes it is contained within. - Example: If a dot is inside a circle AND a square, but NOT a triangle, note this. - Example: If another dot is only inside the triangle, note that too. 2. **List the Conditions for Each Dot:** Create a list of conditions for each dot (e.g., Dot 1: Circle + Square, Dot 2: Triangle only). 3. **Examine Option Figures:** For each option, try to place a dot that satisfies *each* of the conditions from the original figure. 4. **Eliminate Options:** If you cannot find a spot in an option figure for *all* the dots that meets *all* their respective conditions, eliminate that option. #### 2. Key Point - The *relative position* of the dot within the shapes (e.g., top-left of the intersection) is usually not important. Only the *intersection* of shapes is crucial. - You must be able to place *all* dots according to their specific intersection rules in the chosen option. #### 3. Example **Original Figure:** A Venn diagram with a large circle, a square partially overlapping the circle, and a triangle partially overlapping both the circle and the square. - **Dot 1:** Placed in the region common to the Circle and the Square, but not the Triangle. - **Dot 2:** Placed in the region common to the Triangle and the Square, but not the Circle. **Option Figure:** Another Venn diagram. You must find an option where there is a region unique to (Circle + Square - Triangle) AND a region unique to (Triangle + Square - Circle). ### Cubes and Dice This section deals with visualizing 3D objects (cubes and dice) from 2D representations, understanding their faces, and how they relate when unfolded or rotated. #### 1. Standard Dice Rules - Opposite faces sum to 7 (e.g., 1 opposite 6, 2 opposite 5, 3 opposite 4). - This rule applies only to *standard* dice, which will usually be specified. #### 2. Unfolded Cube Patterns - Common patterns for an unfolded cube (net): ``` +---+ | A | +---+---+---+---+ | B | C | D | E | +---+---+---+---+ | F | +---+ ``` - Opposite faces: A & D, B & E, C & F (or similar logic for other nets). - Faces that share an edge cannot be opposite. - **Strategy:** Pick a face, then determine its adjacent and opposite faces by mentally folding the net. #### 3. Rotated Cubes - Given multiple views of the same cube, identify the hidden faces or the pattern on a specific face. - **Strategy:** 1. **Find Common Faces:** Look for two views that share a common face. 2. **Rotate Mentally:** Imagine rotating the cube from one view to another. The faces adjacent to the common face will rotate around it. 3. **Identify Opposites:** If two faces are never seen together, and are always replaced by each other when rotating, they are likely opposite. #### 4. Example (Unfolded Cube) **Net:** ``` +---+---+---+ | 1 | 2 | 3 | +---+---+---+ | 4 | +---+ | 5 | +---+ ``` **Opposite Faces:** - 1 is opposite 3. - 2 is opposite 4. - 5 is opposite the face that would be "above" 1 (if 1-2-3 was the top row). This requires careful mental folding. If 2 is the front, 4 is the bottom, 5 is the back, then 1 is left and 3 is right. *Correction for common net patterns:* In a linear 1-2-3-4-5-6 net: 1 & 3, 2 & 4, 5 & 6 (if 1 is top, 2-3-4-5 are sides, 6 is bottom). *For the given example net:* - If 2 is the front face: 4 is the bottom face. - 1 is the left face, 3 is the right face. - 5 is the top face. - Therefore, 2 is opposite 5. 1 is opposite 3. 4 is opposite the remaining face (which would be 6 if it were a standard net, or the face 'behind' 2). Let's re-evaluate the net: ``` 1 2 3 4 5 ``` - If 3 is the front face. - 2 is left, 4 is right. - 1 is top, 5 is bottom. - Therefore, the pair (2,4), (1,5) and (3, [missing face]). - The given net is slightly ambiguous without a 6th face. Assuming it implicitly means 6 is the face opposite 3. **Common rule for 'cross' pattern:** Alternate faces are opposite. - 1 and 5 are opposite. - 2 and 4 are opposite. - 3 and the "missing" face (or the remaining one if given explicitly) are opposite.