Visual & Logical Reasoning

Overview & Fundamental Concept

Exhaustive AISSEE Class 6 Intelligence guide for Visual & Logical Reasoning. Covers lateral inversion, 11 mirror invariants, 9 water invariants, paper folding/cutting, standard dice opposite laws, figure counting formulas, solved models, and 10 practice drills.

• A Mirror Image is formed by reflection across a Vertical Mirror Line ($XY$) placed to the left or right of an object.
• The Law of Lateral Inversion:
• • Left and Right are REVERSED: The left side of the object appears on the right side of the image, and vice versa.
• • Top and Bottom DO NOT CHANGE: The vertical orientation remains 100% upright.
• • Proximity Principle: Elements closest to the mirror line appear closest to the mirror line in the reflection!
• The 11 Mirror-Invariant Capital Letters:
• • Exactly 11 uppercase English letters possess vertical bilateral symmetry and remain 100% UNCHANGED in a vertical mirror:
• • A, H, I, M, O, T, U, V, W, X, Y (Mnemonic: 'A MOTOUVWXY' or 'A H-I-M in O-T-U-V-W-X-Y').
• • Invariant digits: 0 and 8 (when written without decorative serifs).
• Word Mirror Images: Both the internal letter shapes AND the sequential order of letters invert! The LAST letter of the word becomes the FIRST letter in the mirror image: RED reflects starting with reversed D, then E, then R: $\reflectbox{D}\reflectbox{E}\reflectbox{R}$.

• A Water Image is formed by reflection across a Horizontal Reflection Plane placed underneath the object.
• The Law of Vertical Inversion:
• • Top and Bottom are REVERSED: What is on top flips to the bottom, and what is on the bottom flips to the top.
• • Left and Right DO NOT CHANGE: Lateral orientation remains completely identical!
• • The Word Sequence Principle: Unlike mirror images, the sequence of letters DOES NOT reverse in a water image! The first letter remains the first letter, but is flipped upside down.
• The 9 Water-Invariant Capital Letters:
• • Exactly 9 uppercase letters possess horizontal symmetry and remain 100% UNCHANGED in a water reflection:
• • B, C, D, E, H, I, K, O, X (Mnemonic: 'B-C-D-E HIK-OX': 'Be CD-E, Hike Ox!').
• • Invariant digits: 0, 3, and 8 (when symmetrical).
• • Letters that transform into other letters: M becomes W, and W becomes M!

• 1. Paper Folding and Punching (PFC): A sheet of paper is folded along dotted lines, and holes or cuts are punched into the folded shape.
• • The Reverse-Unfolding Algorithm: Always work backward from the final step to the first step:
• - Step 1: Open the last fold made. Every punched hole reflects across the fold line as a mirror image.
• - Step 2: Open the previous fold. All existing holes reflect across that fold line.
• - The Power of 2 Law: If a paper is folded in half $k$ times, a single punch produces $2^k$ holes! (1 fold = 2 holes; 2 folds = 4 holes; 3 folds = 8 holes).
• 2. Embedded Figures (Hidden Figures): A small target shape is hidden within a complex geometric figure:
• • Rule: The exact shape and line angles of the target figure must be preserved without distortion.
• • Rotation Check: In AISSEE, unless rotation is explicitly prohibited, the hidden figure may appear rotated.

• 1. Cube & Dice Nets:
• • The Standard Dice Law: On a standard dice, opposite faces always sum to 7: $1 \leftrightarrow 6$, $2 \leftrightarrow 5$, $3 \leftrightarrow 4$.
• • The Alternate Face Rule in Open Nets: In any unfolded flat dice net, faces separated by exactly ONE intervening face are OPPOSITE to each other:
• - In a straight strip of 4 squares: Face 1 is opposite Face 3; Face 2 is opposite Face 4.
• - The two projecting wings on the sides are opposite to each other.
• • Adjacent Faces Rule: Two opposite faces can NEVER appear together on any visible view of a folded cube!
• 2. Figure Counting Master Formulas:
• • Counting Triangles in a Divided Rectangle: If a square/rectangle is divided by diagonals into $n$ small triangles: $\text{Total Triangles} = \mathbf{n \times 2}$. (e.g. 8 sectors $\implies 8 \times 2 = 16$ triangles).
• • Counting Triangles in a Segmented Triangle: A triangle divided from one vertex into $n$ vertical strips: $\text{Total Triangles} = \mathbf{\frac{n(n+1)}{2}} = 1 + 2 + 3 + \dots + n$. (e.g. 4 strips $\implies 1+2+3+4 = 10$).
• • Counting Squares in an $n \times n$ Grid: $\text{Total Squares} = 1^2 + 2^2 + 3^2 + \dots + n^2$. (e.g. $3 \times 3$ grid $\implies 1 + 4 + 9 = 14$ squares).