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INTRODUCTION
DICE AND CUBES ARE ESSENTIAL TOPICS IN THE REASONING SECTION OF PSU (PUBLIC SECTOR UNDERTAKINGS) EXAMS. THESE QUESTIONS TEST A CANDIDATE’S SPATIAL VISUALIZATION AND LOGICAL REASONING SKILLS. UNDERSTANDING THE FUNDAMENTAL CONCEPTS AND PRACTICING DIFFERENT TYPES OF PROBLEMS CAN SIGNIFICANTLY ENHANCE ACCURACY AND SPEED.
UNDERSTANDING DICE
A DICE IS A CUBE WITH SIX FACES, EACH DISPLAYING A NUMBER OR SYMBOL. THERE ARE TWO MAJOR TYPES OF DICE USED IN REASONING PROBLEMS:
STANDARD DICE:-
THE SUM OF THE NUMBERS ON OPPOSITE FACES IS ALWAYS 7.
NON-STANDARD DICE:-
NO SUCH RULE APPLIES; THE NUMBERS ON OPPOSITE FACES CAN BE ARBITRARY.
TYPES OF DICE PROBLEMS
IDENTIFYING OPPOSITE FACES:
IF TWO DIFFERENT POSITIONS OF A DICE ARE GIVEN, THE OPPOSITE FACES CAN BE DETERMINED BY COMPARING THE COMMON FACES IN BOTH POSITIONS.
OPEN DICE (UNFOLDED FORM):
A NET OF A CUBE IS GIVEN, AND YOU NEED TO DETERMINE WHICH NUMBER OR SYMBOL APPEARS OPPOSITE TO A GIVEN FACE.
ROTATED DICE:
THE DICE IS ROTATED, AND CANDIDATES MUST FIND THE FACE OPPOSITE TO A GIVEN NUMBER.
COMPARISON DICE PROBLEMS:
DIFFERENT DICE WITH COMMON NUMBERS ARE COMPARED TO INFER THE POSITIONS OF FACES.
SOLVING DICE PROBLEMS
IDENTIFY COMMON NUMBERS BETWEEN DIFFERENT POSITIONS.
USE LOGICAL DEDUCTION TO DETERMINE THE MISSING OPPOSITE FACE.
VISUALIZE THE ROTATION TO MAINTAIN THE RELATIVE POSITIONS OF NUMBERS.
UNDERSTANDING CUBES
A CUBE IS A 3D STRUCTURE WITH SIX FACES, TWELVE EDGES, AND EIGHT VERTICES. QUESTIONS RELATED TO CUBES IN PSU EXAMS TYPICALLY INVOLVE PAINTED CUBES, CUTTING CUBES, OR FINDING PATTERNS.
TYPES OF CUBE PROBLEMS
COUNTING SMALLER CUBES FROM A LARGE CUBE:
IF A LARGE CUBE OF SIDE ‘N’ IS CUT INTO SMALLER CUBES, THE TOTAL NUMBER OF CUBES FORMED IS N³.
CUBES PAINTED ON ONE, TWO, OR THREE FACES:
WHEN A CUBE IS PAINTED AND THEN DIVIDED INTO SMALLER CUBES, CANDIDATES MUST DETERMINE HOW MANY SMALLER CUBES HAVE PAINT ON:
ONE FACE (EDGE CUBES WITHOUT CORNERS)
TWO FACES (EDGE CUBES AT CORNERS)
THREE FACES (CORNER CUBES)
NO FACE (INNER CUBES)
CUBE FOLDING AND UNFOLDING:
CANDIDATES MUST ANALYZE UNFOLDED (NET) CUBES AND PREDICT THE FINAL FOLDED CUBE.
MIRROR AND WATER IMAGES OF CUBES:
DETERMINE HOW A CUBE WOULD APPEAR IN A MIRROR OR UNDER REFLECTION.
SOLVING CUBE PROBLEMS
BREAK DOWN THE PROBLEM INTO SMALLER STEPS.
USE FORMULAS LIKE (N-2)³ FOR CUBES WITH NO PAINT.
OBSERVE SYMMETRY AND PATTERNS IN FOLDED/UNFOLDED CUBES.
FOR CUTTING CUBES, COUNT SYSTEMATICALLY.
TIPS AND TRICKS FOR SOLVING DICE AND CUBE PROBLEMS
PRACTICE VISUALIZATION: DEVELOP THE ABILITY TO MENTALLY ROTATE AND ANALYZE DICE AND CUBE POSITIONS.
MEMORIZE STANDARD PATTERNS: LEARN COMMON NUMBER PLACEMENTS IN DICE AND PAINTED CUBE CASES.
USE ELIMINATION TECHNIQUE: WHEN GIVEN MULTIPLE CHOICES, RULE OUT INCORRECT OPTIONS.
DRAW ROUGH DIAGRAMS: IF NECESSARY, SKETCH ROUGH STRUCTURES FOR BETTER UNDERSTANDING.
SOLVE PREVIOUS YEAR QUESTIONS: THIS WILL HELP UNDERSTAND THE TYPE AND DIFFICULTY OF PSU EXAM QUESTIONS.
CONCLUSION
MASTERING DICE AND CUBES QUESTIONS REQUIRES LOGICAL REASONING AND SPATIAL VISUALIZATION. REGULAR PRACTICE AND UNDERSTANDING OF CONCEPTS WILL SIGNIFICANTLY IMPROVE PROBLEM-SOLVING EFFICIENCY IN PSU EXAMS. KEEP PRACTICING WITH DIFFERENT TYPES OF QUESTIONS TO BUILD CONFIDENCE AND ACCURACY!
SECTION 1: DICE REASONING QUESTIONS
BASIC CONCEPTS OF DICE
- HOW MANY FACES DOES A STANDARD DIE HAVE?
ANSWER: 6
- HOW MANY EDGES DOES A STANDARD DIE HAVE?
ANSWER: 12
- HOW MANY VERTICES DOES A STANDARD DIE HAVE?
ANSWER: 8
- IF ONE FACE OF A DIE SHOWS 5, WHAT IS THE TOTAL SUM OF NUMBERS ON ALL FACES OF A STANDARD DIE?
ANSWER: 21 (SUM OF OPPOSITE FACES IS ALWAYS 7, AND TOTAL IS 1+2+3+4+5+6=21)
- IF A DIE IS NUMBERED FROM 1 TO 6, WHAT IS THE SUM OF NUMBERS ON OPPOSITE FACES?
ANSWER: ALWAYS 7 (1 OPPOSITE TO 6, 2 OPPOSITE TO 5, 3 OPPOSITE TO 4)
DICE ROTATION AND OBSERVATIONS
- A DIE SHOWS 2 ON THE FRONT FACE, 5 ON THE RIGHT FACE, AND 6 ON THE TOP FACE. WHAT NUMBER IS OPPOSITE TO 5?
ANSWER: 1 (SINCE SUM OF OPPOSITE FACES IS 7)
- A DIE HAS NUMBERS 1 TO 6 ON ITS FACES. IF THE TOP FACE SHOWS 4 AND ADJACENT FACES SHOW 2, 3, 5, WHAT NUMBER IS ON THE BOTTOM FACE?
ANSWER: 7 – 4 = 3
- IF A DIE IS ROLLED AND THE FACE SHOWING 3 MOVES TO THE TOP, WHAT HAPPENS TO THE NUMBER ON THE OPPOSITE FACE?
ANSWER: IT MOVES TO THE BOTTOM.
- A CUBE WITH DIFFERENT NUMBERS ON EACH FACE IS ROTATED TWICE. THE NUMBERS 2, 3, AND 6 REMAIN VISIBLE IN DIFFERENT POSITIONS. WHICH NUMBER WAS ORIGINALLY AT THE BOTTOM?
ANSWER: THE MISSING NUMBER FROM THE SET 1-6.
- TWO DICE ARE PLACED SIDE BY SIDE SHOWING NUMBERS 1, 3, 5, AND 2, 6, 4. WHAT IS THE SUM OF HIDDEN NUMBERS?
ANSWER: 7 + 7 = 14
SECTION 2: CUBE REASONING QUESTIONS
BASIC CUBE CONCEPTS
- HOW MANY FACES DOES A CUBE HAVE?
ANSWER: 6
- HOW MANY EDGES DOES A CUBE HAVE?
ANSWER: 12
- HOW MANY VERTICES DOES A CUBE HAVE?
ANSWER: 8
- A CUBE IS PAINTED ON ALL ITS SIDES AND THEN CUT INTO 64 SMALLER CUBES. HOW MANY CUBES WILL HAVE THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES)
- HOW MANY CUBES WILL HAVE TWO FACES PAINTED IN THE PREVIOUS QUESTION?
ANSWER: 24 (EDGE CUBES, EXCLUDING CORNERS)
CUTTING AND PAINTING CUBES
- A CUBE IS PAINTED AND CUT INTO 27 SMALLER CUBES. HOW MANY CUBES WILL HAVE ONLY ONE FACE PAINTED?
ANSWER: 6 (CENTER FACE CUBES)
- HOW MANY SMALLER CUBES WILL HAVE NO PAINT ON THEM IN A 3×3×3 CUBE?
ANSWER: 1 (THE CENTER CUBE)
- IF A 4×4×4 CUBE IS PAINTED ON ALL SIDES AND CUT INTO SMALLER CUBES, HOW MANY WILL HAVE ONLY ONE FACE PAINTED?
ANSWER: 24
- A CUBE OF 5×5×5 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY CUBES WILL HAVE THREE FACES PAINTED?
ANSWER: 8
- HOW MANY CUBES WILL HAVE NO FACE PAINTED IN A 5×5×5 CUBE?
ANSWER: 27 (INNER 3×3×3 CUBE)
SECTION 3: ADVANCED DICE AND CUBE PROBLEMS
DICE COMBINATIONS AND LOGICAL REASONING
- TWO DICE ARE THROWN TOGETHER. WHAT IS THE PROBABILITY OF GETTING A SUM OF 7?
ANSWER: 6/36 = 1/6
- IF THREE DICE ARE ROLLED TOGETHER, HOW MANY DIFFERENT SUMS ARE POSSIBLE?
ANSWER: 16 (FROM 3 TO 18)
- IF TWO OPPOSITE FACES OF A DICE SHOW 6 AND 1, WHICH PAIR OF NUMBERS CANNOT APPEAR ON ADJACENT FACES?
ANSWER: 6 AND 1
- A DICE IS THROWN TWICE. IF THE FIRST THROW SHOWS 3, WHAT IS THE PROBABILITY OF GETTING AN EVEN NUMBER IN THE SECOND THROW?
ANSWER: 3/6 = 1/2
- A DIE IS ROLLED THREE TIMES. WHAT IS THE PROBABILITY THAT ALL THREE ROLLS SHOW DIFFERENT NUMBERS?
ANSWER: (6/6) × (5/6) × (4/6) = 20/36
SECTION 3: ADVANCED DICE AND CUBE PROBLEMS (CONTINUED)
DICE LOGICAL REASONING QUESTIONS
- A STANDARD DIE IS ROLLED. WHAT IS THE PROBABILITY OF ROLLING A NUMBER GREATER THAN 4?
ANSWER: 2/6 = 1/3 (NUMBERS GREATER THAN 4 ARE 5 AND 6)
- TWO DICE ARE THROWN TOGETHER. WHAT IS THE PROBABILITY OF GETTING A SUM OF 8?
ANSWER: 5/36 (POSSIBLE PAIRS: (2,6), (3,5), (4,4), (5,3), (6,2))
- TWO DICE ARE THROWN TOGETHER. WHAT IS THE PROBABILITY OF GETTING AT LEAST ONE 6?
ANSWER: 11/36 (TOTAL OUTCOMES: 36; FAVORABLE OUTCOMES: 11)
- IF A DICE IS THROWN TWICE, WHAT IS THE PROBABILITY OF GETTING THE SAME NUMBER BOTH TIMES?
ANSWER: 1/6 (ONLY 6 FAVORABLE OUTCOMES: (1,1), (2,2), …, (6,6))
- IF A FAIR DIE IS ROLLED, WHAT IS THE PROBABILITY THAT THE RESULT IS AN ODD NUMBER?
ANSWER: 3/6 = 1/2 (ODD NUMBERS ARE 1, 3, AND 5)
CUBE COUNTING AND PAINTING
- A CUBE IS PAINTED ON ALL SIX FACES AND THEN CUT INTO 64 SMALLER CUBES. HOW MANY CUBES WILL HAVE EXACTLY TWO FACES PAINTED?
ANSWER: 24 (EDGE CUBES)
- A CUBE IS PAINTED AND THEN CUT INTO 125 SMALLER CUBES. HOW MANY WILL HAVE EXACTLY THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES)
- A CUBE IS PAINTED AND THEN CUT INTO 125 SMALLER CUBES. HOW MANY WILL HAVE EXACTLY ONE FACE PAINTED?
ANSWER: 54 (FACE CENTER CUBES)
- A CUBE OF 10×10×10 IS PAINTED AND CUT INTO 1000 SMALLER CUBES. HOW MANY WILL HAVE NO PAINT AT ALL?
ANSWER: 512 (INNER 8×8×8 CUBE)
- A CUBE IS DIVIDED INTO 64 SMALLER CUBES. HOW MANY CUBES HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 56 (TOTAL – INNER CUBES)
DICE NUMBER ARRANGEMENT
- IN A DICE, THE NUMBERS ON ADJACENT FACES ARE 3, 5, AND 2. WHAT IS THE NUMBER ON THE OPPOSITE FACE OF 3?
ANSWER: 4 (SINCE SUM OF OPPOSITE FACES IS 7)
- A STANDARD DIE IS MODIFIED SO THAT OPPOSITE FACES SUM TO 8 INSTEAD OF 7. IF 6 IS OPPOSITE TO 2, WHAT NUMBER IS OPPOSITE TO 3?
ANSWER: 5 (PAIRS ARE (1,7), (2,6), (3,5), (4,4))
- A DIE IS SHOWN WITH 2 ON THE FRONT, 3 ON THE RIGHT, AND 5 ON THE TOP. WHAT NUMBER IS OPPOSITE TO 5?
ANSWER: 1 (SUM OF OPPOSITE FACES IS 7)
- A CUBE HAS NUMBERS 1 TO 6 WRITTEN ON ITS FACES. IF THE SUM OF TWO OPPOSITE FACES IS ALWAYS 7, WHICH NUMBERS CANNOT BE ADJACENT?
ANSWER: 1 AND 6, 2 AND 5, 3 AND 4
- IF A DIE IS ROLLED AND THE FACE SHOWING 5 MOVES TO THE TOP, WHICH FACE WAS ORIGINALLY ON TOP?
- ANSWER: THE FACE OPPOSITE TO 5
ADVANCED CUBE PROBLEMS
- A CUBE IS CUT INTO 216 SMALLER CUBES. HOW MANY WILL HAVE NO FACE PAINTED?
ANSWER: 64 (INNER 4×4×4 CUBE)
- A CUBE IS PAINTED AND THEN CUT INTO 343 SMALLER CUBES. HOW MANY WILL HAVE EXACTLY ONE FACE PAINTED?
ANSWER: 150
- A CUBE IS PAINTED AND THEN CUT INTO 512 SMALLER CUBES. HOW MANY WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 488
- A CUBE IS PAINTED ON ONLY TWO OPPOSITE FACES AND THEN CUT INTO 125 SMALLER CUBES. HOW MANY CUBES WILL HAVE NO PAINT?
ANSWER: 81 (INNER 3×3×3 CUBE IN THE MIDDLE LAYERS)
- A CUBE IS DIVIDED INTO 1000 SMALLER CUBES. HOW MANY OF THESE HAVE ALL THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES)
MORE DICE PROBABILITY AND LOGICAL REASONING
- IF A FAIR DIE IS ROLLED, WHAT IS THE PROBABILITY OF GETTING A PRIME NUMBER?
ANSWER: 1/2 (PRIME NUMBERS: 2, 3, 5)
- A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY THAT THE SUM IS AT MOST 4?
ANSWER: 3/36 = 1/12 (POSSIBLE PAIRS: (1,1), (1,2), (2,1))
- A DIE IS THROWN. WHAT IS THE PROBABILITY OF GETTING AN EVEN NUMBER OR A NUMBER GREATER THAN 4?
ANSWER: 2/3 (EVEN: 2, 4, 6; GREATER THAN 4: 5, 6)
- A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY THAT AT LEAST ONE ROLL RESULTS IN A 6?
ANSWER: 11/36 (COMPLEMENT OF GETTING NO 6)
- A DIE IS ROLLED THREE TIMES. WHAT IS THE PROBABILITY OF GETTING AT LEAST ONE 5?
ANSWER: 91/216
SECTION 4: DICE PROBABILITY AND LOGICAL REASONING (CONTINUED)
PROBABILITY-BASED DICE QUESTIONS
- A DIE IS ROLLED ONCE. WHAT IS THE PROBABILITY OF GETTING A NUMBER LESS THAN 4?
ANSWER: 3/6 = 1/2 (NUMBERS LESS THAN 4 ARE 1, 2, 3)
- IF TWO DICE ARE THROWN TOGETHER, WHAT IS THE PROBABILITY OF GETTING A SUM OF 9?
ANSWER: 4/36 = 1/9 (POSSIBLE PAIRS: (3,6), (4,5), (5,4), (6,3))
- A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY THAT THE FIRST ROLL IS A 4 AND THE SECOND ROLL IS AN EVEN NUMBER?
ANSWER: (1/6) × (3/6) = 1/12
- WHAT IS THE PROBABILITY OF ROLLING TWO DICE AND GETTING AT LEAST ONE 1?
ANSWER: 11/36 (COMPLEMENT METHOD: PROBABILITY OF NO 1 IN BOTH ROLLS = 25/36, SO 1 – 25/36 = 11/36)
- THREE DICE ARE ROLLED. WHAT IS THE PROBABILITY THAT ALL THREE NUMBERS ARE DIFFERENT?
ANSWER: (6/6) × (5/6) × (4/6) = 20/36
DICE ARRANGEMENTS & OPPOSITE FACES
- A DIE IS PLACED SUCH THAT THE NUMBER 3 IS ON TOP AND 5 IS ON THE FRONT. WHAT NUMBER IS OPPOSITE TO 3?
ANSWER: 4 (SINCE THE SUM OF OPPOSITE FACES IS 7)
- A STANDARD DIE IS ROLLED, AND 2 APPEARS ON THE FRONT WHILE 6 APPEARS ON THE TOP. WHAT NUMBER IS OPPOSITE TO 6?
ANSWER: 1
- A DICE HAS NUMBERS ARRANGED SUCH THAT OPPOSITE FACES ADD UP TO 7. IF ONE DIE IS ROTATED AND 1 IS AT THE TOP, WHAT NUMBER MUST BE AT THE BOTTOM?
ANSWER: 6
- A DICE HAS ADJACENT FACES SHOWING 2, 4, AND 6. WHAT NUMBER IS OPPOSITE TO 2?
ANSWER: 5
- A DIE IS ROLLED, AND ITS THREE VISIBLE FACES ARE 1, 3, AND 5. WHAT IS THE SUM OF THE HIDDEN FACES?
ANSWER: 9 (TOTAL SUM IS 21, SUM OF VISIBLE FACES IS 1+3+5 = 9, SO HIDDEN SUM = 21 – 9 = 12)
SECTION 5: CUBE CUTTING AND PAINTING
BASIC CUBE CONCEPTS
- A CUBE HAS 6 FACES AND IS PAINTED ON ALL SIDES. IF IT IS CUT INTO 27 SMALLER CUBES, HOW MANY HAVE THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES)
- A CUBE OF 4×4×4 IS PAINTED ON ALL FACES AND CUT INTO SMALLER CUBES. HOW MANY CUBES HAVE TWO FACES PAINTED?
ANSWER: 24 (EDGE CUBES, EXCLUDING CORNERS)
- A CUBE OF 3×3×3 IS CUT INTO 27 SMALLER CUBES. HOW MANY OF THEM WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 26 (ONLY THE CENTER CUBE REMAINS UNPAINTED)
- A CUBE IS PAINTED AND THEN CUT INTO 125 SMALLER CUBES. HOW MANY WILL HAVE EXACTLY ONE FACE PAINTED?
ANSWER: 54 (FACE-CENTER CUBES)
- A CUBE OF 5×5×5 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY WILL HAVE ALL THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES)
ADVANCED CUBE QUESTIONS
- A CUBE IS CUT INTO 64 SMALLER CUBES. HOW MANY WILL HAVE NO PAINT AT ALL?
ANSWER: 8 (INNER 2×2×2 CUBE)
- A CUBE IS DIVIDED INTO 125 SMALLER CUBES. HOW MANY WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 98 (TOTAL CUBES – UNPAINTED CUBES)
- A CUBE OF 6×6×6 IS PAINTED ON ALL FACES AND THEN CUT INTO SMALLER CUBES. HOW MANY CUBES WILL HAVE TWO FACES PAINTED?
ANSWER: 48 (EDGE CUBES, EXCLUDING CORNERS)
- A CUBE IS PAINTED ON ONLY TWO OPPOSITE FACES AND THEN CUT INTO 64 SMALLER CUBES. HOW MANY CUBES WILL HAVE NO PAINT AT ALL?
ANSWER: 32 (INNER CUBES THAT DO NOT TOUCH THE PAINTED FACES)
- A CUBE IS PAINTED ON ONLY THREE ADJACENT FACES AND THEN CUT INTO 125 SMALLER CUBES. HOW MANY CUBES WILL HAVE NO PAINT AT ALL?
ANSWER: 64 (INNER 4×4×4 CUBE)
SECTION 6: MIXED DICE AND CUBE PUZZLES
DICE AND CUBE LOGICAL REASONING
- A CUBE IS PLACED IN SUCH A WAY THAT THREE OF ITS ADJACENT FACES SHOW NUMBERS 1, 2, AND 3. WHAT NUMBER IS OPPOSITE TO 1?
ANSWER: 6 (SINCE SUM OF OPPOSITE FACES IS 7)
- A CUBE IS CUT INTO 216 SMALLER CUBES. HOW MANY WILL HAVE THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES)
- A CUBE OF 7×7×7 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY CUBES WILL HAVE NO PAINT AT ALL?
ANSWER: 125 (INNER 5×5×5 CUBE)
- A CUBE OF 8×8×8 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY CUBES WILL HAVE EXACTLY ONE FACE PAINTED?
ANSWER: 96 (FACE-CENTER CUBES)
- A CUBE IS CUT INTO 512 SMALLER CUBES. HOW MANY WILL HAVE NO PAINT AT ALL?
ANSWER: 216 (INNER 6×6×6 CUBE)
SECTION 7: ADVANCED DICE & CUBE PROBABILITY
- IF TWO DICE ARE ROLLED, WHAT IS THE PROBABILITY OF GETTING A SUM OF 10?
ANSWER: 3/36 = 1/12 (POSSIBLE PAIRS: (4,6), (5,5), (6,4))
- A DICE IS THROWN TWICE. WHAT IS THE PROBABILITY THAT BOTH THROWS GIVE PRIME NUMBERS?
ANSWER: (3/6) × (3/6) = 9/36 = 1/4
- IF THREE DICE ARE ROLLED, WHAT IS THE PROBABILITY OF GETTING AT LEAST ONE 6?
ANSWER: 91/216
- A DIE IS ROLLED. WHAT IS THE PROBABILITY OF GETTING AN EVEN NUMBER OR A NUMBER GREATER THAN 4?
ANSWER: 2/3 (EVEN NUMBERS: 2, 4, 6; GREATER THAN 4: 5, 6)
- A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY OF GETTING A 4 AT LEAST ONCE?
ANSWER: 11/36
SECTION 8: ADVANCED DICE AND CUBE REASONING (CONTINUED)
DICE PROBABILITY AND LOGICAL QUESTIONS
- A DIE IS ROLLED ONCE. WHAT IS THE PROBABILITY OF GETTING A NUMBER GREATER THAN 2?
ANSWER: 4/6 = 2/3 (NUMBERS GREATER THAN 2 ARE 3, 4, 5, 6)
- IF TWO DICE ARE ROLLED TOGETHER, WHAT IS THE PROBABILITY OF GETTING AN EVEN SUM?
- ANSWER: 18/36 = 1/2 (HALF OF ALL POSSIBLE SUMS ARE EVEN)
- A DIE IS THROWN. WHAT IS THE PROBABILITY OF GETTING A NUMBER THAT IS EITHER A MULTIPLE OF 2 OR 3?
ANSWER: 4/6 = 2/3 (MULTIPLES OF 2: 2, 4, 6; MULTIPLES OF 3: 3, 6)
- TWO DICE ARE THROWN TOGETHER. WHAT IS THE PROBABILITY OF GETTING A SUM OF 11?
ANSWER: 2/36 = 1/18 (POSSIBLE PAIRS: (5,6), (6,5))
- THREE DICE ARE ROLLED. WHAT IS THE PROBABILITY THAT ALL THREE NUMBERS ARE THE SAME?
ANSWER: 6/216 = 1/36 (SIX POSSIBLE OUTCOMES: (1,1,1), (2,2,2), …, (6,6,6))
DICE OPPOSITE FACES AND ARRANGEMENTS
- A STANDARD DIE IS PLACED SO THAT 4 IS ON TOP AND 2 IS IN FRONT. WHAT NUMBER IS OPPOSITE TO 4?
ANSWER: 3 (SINCE OPPOSITE FACES SUM TO 7)
- A DIE SHOWS 5 ON THE FRONT, 3 ON THE TOP, AND 2 ON THE RIGHT. WHAT NUMBER IS OPPOSITE TO 5?
ANSWER: 2
- A CUBE IS PLACED SUCH THAT 1, 3, AND 6 ARE VISIBLE. WHAT IS THE NUMBER OPPOSITE TO 6?
ANSWER: 1
- A DIE IS POSITIONED SO THAT THE SUM OF ADJACENT FACES IS ALWAYS EVEN. WHICH NUMBER MUST BE OPPOSITE TO 6?
ANSWER: 2 (ENSURES THAT OPPOSITE FACES SUM TO AN EVEN NUMBER)
- A DIE IS POSITIONED SUCH THAT 3 IS AT THE TOP, 2 IS AT THE FRONT, AND 5 IS AT THE RIGHT. WHAT NUMBER IS AT THE BACK?
ANSWER: 6
CUBE CUTTING AND PAINTING QUESTIONS
- A CUBE IS PAINTED AND THEN CUT INTO 216 SMALLER CUBES. HOW MANY CUBES HAVE THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES)
- A CUBE OF 6×6×6 IS CUT INTO SMALLER CUBES. HOW MANY WILL HAVE EXACTLY ONE FACE PAINTED?
ANSWER: 96 (FACE-CENTER CUBES)
- A CUBE OF 5×5×5 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY CUBES HAVE TWO FACES PAINTED?
ANSWER: 48 (EDGE CUBES, EXCLUDING CORNERS)
- A CUBE IS DIVIDED INTO 343 SMALLER CUBES. HOW MANY WILL HAVE NO PAINT AT ALL?
ANSWER: 125 (INNER 5×5×5 CUBE)
- A CUBE IS PAINTED ON ALL FACES AND THEN CUT INTO 512 SMALLER CUBES. HOW MANY CUBES WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 488 (TOTAL CUBES – UNPAINTED CUBES)
MIXED DICE AND CUBE PUZZLES
- A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY OF GETTING A SUM OF 8?
ANSWER: 5/36 (POSSIBLE PAIRS: (2,6), (3,5), (4,4), (5,3), (6,2))
- A DIE IS THROWN THREE TIMES. WHAT IS THE PROBABILITY OF GETTING AT LEAST ONE 1?
ANSWER: 91/216
- IF A DIE IS ROLLED TWICE, WHAT IS THE PROBABILITY OF GETTING THE SAME NUMBER BOTH TIMES?
ANSWER: 1/6
- A CUBE IS PAINTED AND CUT INTO 64 SMALLER CUBES. HOW MANY WILL HAVE EXACTLY TWO FACES PAINTED?
ANSWER: 24
- A DIE IS PLACED SO THAT THE NUMBERS 1, 3, AND 5 ARE VISIBLE. WHAT IS THE SUM OF THE HIDDEN NUMBERS?
ANSWER: 12 (TOTAL SUM IS 21, VISIBLE SUM IS 9, SO HIDDEN SUM = 21 – 9 = 12)
SECTION 9: ADVANCED DICE AND CUBE REASONING (CONTINUED)
DICE PROBABILITY AND LOGICAL QUESTIONS
- A DIE IS ROLLED ONCE. WHAT IS THE PROBABILITY OF GETTING AN ODD NUMBER?
ANSWER: 3/6 = 1/2 (ODD NUMBERS: 1, 3, 5) - TWO DICE ARE ROLLED TOGETHER. WHAT IS THE PROBABILITY OF GETTING A SUM OF 7?
ANSWER: 6/36 = 1/6 (POSSIBLE PAIRS: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)) - A DIE IS THROWN TWICE. WHAT IS THE PROBABILITY OF GETTING TWO DIFFERENT NUMBERS?
ANSWER: 5/6 (TOTAL CASES = 36, SAME NUMBER CASES = 6, SO DIFFERENT CASES = 30/36 = 5/6) - A DIE IS THROWN. WHAT IS THE PROBABILITY OF GETTING A NUMBER DIVISIBLE BY 2 OR 3?
ANSWER: 4/6 = 2/3 (NUMBERS: 2, 3, 4, 6) - IF A DIE IS ROLLED TWICE, WHAT IS THE PROBABILITY OF GETTING AT LEAST ONE EVEN NUMBER?
– ANSWER: 3/4 (COMPLEMENT METHOD: PROBABILITY OF GETTING NO EVEN NUMBER = 1/4, SO 1 – 1/4 = 3/4)
DICE OPPOSITE FACES AND ARRANGEMENTS
- A STANDARD DIE IS PLACED SO THAT 3 IS ON TOP AND 2 IS IN FRONT. WHAT NUMBER IS OPPOSITE TO 3?
ANSWER: 4 (SINCE OPPOSITE FACES SUM TO 7) - A DIE HAS THE NUMBERS 2, 5, AND 6 ON ADJACENT FACES. WHAT IS THE NUMBER OPPOSITE TO 2?
ANSWER: 5 - A CUBE IS PLACED SUCH THAT 1, 4, AND 6 ARE VISIBLE. WHAT NUMBER IS OPPOSITE TO 1?
ANSWER: 6 - A DIE IS POSITIONED SO THAT THE SUM OF ADJACENT FACES IS ALWAYS AN EVEN NUMBER. WHICH NUMBER MUST BE OPPOSITE TO 3?
ANSWER: 5 - A DIE IS POSITIONED SUCH THAT 2 IS AT THE TOP, 4 IS AT THE FRONT, AND 5 IS AT THE RIGHT. WHAT NUMBER IS AT THE BACK?
ANSWER: 3
CUBE CUTTING AND PAINTING QUESTIONS
- A CUBE IS PAINTED ON ALL FACES AND CUT INTO 64 SMALLER CUBES. HOW MANY CUBES HAVE THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES) - A CUBE OF 5×5×5 IS PAINTED ON ALL SIX FACES AND CUT INTO SMALLER CUBES. HOW MANY WILL HAVE TWO FACES PAINTED?
ANSWER: 48 (EDGE CUBES) - A CUBE OF 6×6×6 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY CUBES HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 152 - A CUBE OF 4×4×4 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY WILL HAVE EXACTLY ONE FACE PAINTED?
ANSWER: 24 - A CUBE IS DIVIDED INTO 216 SMALLER CUBES. HOW MANY CUBES WILL HAVE NO PAINT AT ALL?
ANSWER: 64 (INNER 4×4×4 CUBE)
MIXED DICE AND CUBE PUZZLES
- A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY OF GETTING A SUM OF 6?
ANSWER: 5/36 (POSSIBLE PAIRS: (1,5), (2,4), (3,3), (4,2), (5,1)) - A DIE IS THROWN THREE TIMES. WHAT IS THE PROBABILITY OF GETTING AT LEAST ONE ODD NUMBER?
ANSWER: 7/8 - A CUBE IS PAINTED AND THEN CUT INTO 125 SMALLER CUBES. HOW MANY WILL HAVE EXACTLY TWO FACES PAINTED?
ANSWER: 48 - A CUBE IS PAINTED AND THEN CUT INTO 512 SMALLER CUBES. HOW MANY WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 488 - A DIE IS PLACED SO THAT THE NUMBERS 2, 4, AND 6 ARE VISIBLE. WHAT IS THE SUM OF THE HIDDEN NUMBERS?
ANSWER: 9 (TOTAL SUM = 21, VISIBLE SUM = 12, SO HIDDEN SUM = 9)
SECTION 10: ADVANCED DICE AND CUBE REASONING (CONTINUED)
DICE PROBABILITY AND LOGICAL QUESTIONS
- A DIE IS ROLLED ONCE. WHAT IS THE PROBABILITY OF GETTING A PRIME NUMBER?
ANSWER: 3/6 = 1/2 (PRIME NUMBERS: 2, 3, 5) - TWO DICE ARE ROLLED TOGETHER. WHAT IS THE PROBABILITY OF GETTING A SUM OF 4?
ANSWER: 3/36 = 1/12 (POSSIBLE PAIRS: (1,3), (2,2), (3,1)) - IF A DIE IS ROLLED TWICE, WHAT IS THE PROBABILITY THAT THE FIRST ROLL IS AN ODD NUMBER AND THE SECOND ROLL IS AN EVEN NUMBER?
ANSWER: (3/6) × (3/6) = 1/4 - A DIE IS THROWN. WHAT IS THE PROBABILITY OF GETTING A NUMBER GREATER THAN 3?
ANSWER: 3/6 = 1/2 (NUMBERS: 4, 5, 6) - A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY THAT BOTH NUMBERS ARE THE SAME?
ANSWER: 6/36 = 1/6
DICE OPPOSITE FACES AND ARRANGEMENTS
- A STANDARD DIE IS PLACED SUCH THAT 5 IS ON TOP AND 3 IS IN FRONT. WHAT NUMBER IS OPPOSITE TO 5?
ANSWER: 2 (SINCE OPPOSITE FACES SUM TO 7) - A DIE HAS NUMBERS 1, 4, AND 6 ON ADJACENT FACES. WHAT NUMBER IS OPPOSITE TO 1?
ANSWER: 6 - A CUBE IS PLACED SUCH THAT 3, 5, AND 2 ARE VISIBLE. WHAT NUMBER IS OPPOSITE TO 2?
ANSWER: 4 - A DIE IS POSITIONED SO THAT THE SUM OF ADJACENT FACES IS ALWAYS ODD. WHICH NUMBER MUST BE OPPOSITE TO 4?
ANSWER: 3 - A DIE IS POSITIONED SUCH THAT 1 IS AT THE TOP, 3 IS AT THE FRONT, AND 5 IS AT THE RIGHT. WHAT NUMBER IS AT THE BOTTOM?
ANSWER: 6
CUBE CUTTING AND PAINTING QUESTIONS
- A CUBE IS PAINTED ON ALL FACES AND CUT INTO 125 SMALLER CUBES. HOW MANY CUBES HAVE THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES) - A CUBE OF 5×5×5 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY WILL HAVE EXACTLY TWO FACES PAINTED?
ANSWER: 48 (EDGE CUBES) - A CUBE OF 6×6×6 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY WILL HAVE EXACTLY ONE FACE PAINTED?
ANSWER: 96 (FACE-CENTER CUBES) - A CUBE IS DIVIDED INTO 343 SMALLER CUBES. HOW MANY WILL HAVE NO PAINT AT ALL?
ANSWER: 125 (INNER 5×5×5 CUBE) - A CUBE IS PAINTED ON ALL FACES AND THEN CUT INTO 512 SMALLER CUBES. HOW MANY CUBES WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 488
MIXED DICE AND CUBE PUZZLES
- A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY OF GETTING A SUM OF 9?
ANSWER: 4/36 = 1/9 (POSSIBLE PAIRS: (3,6), (4,5), (5,4), (6,3)) - A DIE IS THROWN THREE TIMES. WHAT IS THE PROBABILITY OF GETTING AT LEAST ONE EVEN NUMBER?
– ANSWER: 7/8 - A CUBE IS PAINTED AND THEN CUT INTO 64 SMALLER CUBES. HOW MANY WILL HAVE EXACTLY TWO FACES PAINTED?
ANSWER: 24 - A CUBE IS PAINTED AND THEN CUT INTO 216 SMALLER CUBES. HOW MANY WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 152 - A DIE IS PLACED SO THAT THE NUMBERS 2, 4, AND 6 ARE VISIBLE. WHAT IS THE SUM OF THE HIDDEN NUMBERS?
ANSWER: 9 (TOTAL SUM = 21, VISIBLE SUM = 12, SO HIDDEN SUM = 9)
SECTION 11: ADVANCED DICE AND CUBE REASONING (CONTINUED)
DICE PROBABILITY AND LOGICAL QUESTIONS
- A DIE IS ROLLED ONCE. WHAT IS THE PROBABILITY OF GETTING A NUMBER LESS THAN 4?
ANSWER: 3/6 = 1/2 (NUMBERS: 1, 2, 3) - TWO DICE ARE ROLLED TOGETHER. WHAT IS THE PROBABILITY OF GETTING A SUM OF 5?
ANSWER: 4/36 = 1/9 (POSSIBLE PAIRS: (1,4), (2,3), (3,2), (4,1)) - IF A DIE IS ROLLED TWICE, WHAT IS THE PROBABILITY THAT THE SECOND ROLL IS GREATER THAN THE FIRST?
ANSWER: 15/36 = 5/12 (COUNT ALL VALID PAIRS) - A DIE IS THROWN. WHAT IS THE PROBABILITY OF GETTING A NUMBER DIVISIBLE BY 3?
ANSWER: 2/6 = 1/3 (NUMBERS: 3, 6) - A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY THAT AT LEAST ONE ROLL RESULTS IN A 6?
ANSWER: 11/36 (USE COMPLEMENT METHOD: 1 – (5/6 × 5/6))
DICE OPPOSITE FACES AND ARRANGEMENTS
- A STANDARD DIE IS PLACED SUCH THAT 2 IS ON TOP AND 5 IS IN FRONT. WHAT NUMBER IS OPPOSITE TO 2?
ANSWER: 5 (OPPOSITE FACES SUM TO 7) - A DIE HAS NUMBERS 3, 4, AND 5 ON ADJACENT FACES. WHAT NUMBER IS OPPOSITE TO 3?
ANSWER: 6 - A CUBE IS PLACED SUCH THAT 2, 4, AND 6 ARE VISIBLE. WHAT NUMBER IS OPPOSITE TO 2?
ANSWER: 5 - A DIE IS POSITIONED SO THAT THE SUM OF ADJACENT FACES IS ALWAYS AN ODD NUMBER. WHICH NUMBER MUST BE OPPOSITE TO 2?
ANSWER: 5 - A DIE IS POSITIONED SUCH THAT 1 IS AT THE TOP, 3 IS AT THE FRONT, AND 6 IS AT THE RIGHT. WHAT NUMBER IS AT THE BOTTOM?
ANSWER: 2
CUBE CUTTING AND PAINTING QUESTIONS
- A CUBE IS PAINTED ON ALL FACES AND CUT INTO 216 SMALLER CUBES. HOW MANY CUBES HAVE THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES) - A CUBE OF 7×7×7 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY WILL HAVE EXACTLY TWO FACES PAINTED?
ANSWER: 72 (EDGE CUBES) - A CUBE OF 6×6×6 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY WILL HAVE EXACTLY ONE FACE PAINTED?
ANSWER: 96 (FACE-CENTER CUBES) - A CUBE IS DIVIDED INTO 512 SMALLER CUBES. HOW MANY WILL HAVE NO PAINT AT ALL?
ANSWER: 216 (INNER 6×6×6 CUBE) - A CUBE IS PAINTED ON ALL FACES AND THEN CUT INTO 729 SMALLER CUBES. HOW MANY CUBES WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 665
MIXED DICE AND CUBE PUZZLES
- A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY OF GETTING A SUM OF 10?
ANSWER: 3/36 = 1/12 (POSSIBLE PAIRS: (4,6), (5,5), (6,4)) - A DIE IS THROWN THREE TIMES. WHAT IS THE PROBABILITY OF GETTING AT LEAST ONE PRIME NUMBER?
ANSWER: 19/27 - A CUBE IS PAINTED AND THEN CUT INTO 64 SMALLER CUBES. HOW MANY WILL HAVE EXACTLY TWO FACES PAINTED?
ANSWER: 24 - A CUBE IS PAINTED AND THEN CUT INTO 216 SMALLER CUBES. HOW MANY WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 152 - A DIE IS PLACED SO THAT THE NUMBERS 2, 3, AND 5 ARE VISIBLE. WHAT IS THE SUM OF THE HIDDEN NUMBERS?
ANSWER: 11 (TOTAL SUM = 21, VISIBLE SUM = 10, SO HIDDEN SUM = 11)
SECTION 12: ADVANCED DICE AND CUBE REASONING (CONTINUED)
DICE PROBABILITY AND LOGICAL QUESTIONS
- A DIE IS ROLLED ONCE. WHAT IS THE PROBABILITY OF GETTING A NUMBER GREATER THAN 4?
ANSWER: 2/6 = 1/3 (NUMBERS: 5, 6) - TWO DICE ARE ROLLED TOGETHER. WHAT IS THE PROBABILITY OF GETTING A SUM OF 3?
ANSWER: 2/36 = 1/18 (POSSIBLE PAIRS: (1,2), (2,1)) - IF A DIE IS ROLLED TWICE, WHAT IS THE PROBABILITY THAT THE SECOND ROLL IS LESS THAN THE FIRST?
ANSWER: 10/36 = 5/18 (COUNT ALL VALID PAIRS) - A DIE IS THROWN. WHAT IS THE PROBABILITY OF GETTING AN EVEN NUMBER?
ANSWER: 3/6 = 1/2 (NUMBERS: 2, 4, 6) - A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY THAT AT LEAST ONE ROLL RESULTS IN A 1?
–ANSWER: 11/36 (USE COMPLEMENT METHOD: 1 – (5/6 × 5/6))
DICE OPPOSITE FACES AND ARRANGEMENTS
- A STANDARD DIE IS PLACED SUCH THAT 6 IS ON TOP AND 2 IS IN FRONT. WHAT NUMBER IS OPPOSITE TO 6?
ANSWER: 1 (SINCE OPPOSITE FACES SUM TO 7) - A DIE HAS NUMBERS 1, 3, AND 5 ON ADJACENT FACES. WHAT NUMBER IS OPPOSITE TO 1?
ANSWER: 6 - A CUBE IS PLACED SUCH THAT 4, 5, AND 6 ARE VISIBLE. WHAT NUMBER IS OPPOSITE TO 4?
ANSWER: 3 - A DIE IS POSITIONED SO THAT THE SUM OF ADJACENT FACES IS ALWAYS ODD. WHICH NUMBER MUST BE OPPOSITE TO 3?
– ANSWER: 4 - A DIE IS POSITIONED SUCH THAT 1 IS AT THE TOP, 2 IS AT THE FRONT, AND 3 IS AT THE RIGHT. WHAT NUMBER IS AT THE BOTTOM?
ANSWER: 6
CUBE CUTTING AND PAINTING QUESTIONS
- A CUBE IS PAINTED ON ALL FACES AND CUT INTO 512 SMALLER CUBES. HOW MANY CUBES HAVE THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES) - A CUBE OF 8×8×8 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY WILL HAVE EXACTLY TWO FACES PAINTED?
ANSWER: 96 (EDGE CUBES) - A CUBE OF 7×7×7 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY WILL HAVE EXACTLY ONE FACE PAINTED?
ANSWER: 150 (FACE-CENTER CUBES) - A CUBE IS DIVIDED INTO 1000 SMALLER CUBES. HOW MANY WILL HAVE NO PAINT AT ALL?
ANSWER: 512 (INNER 8×8×8 CUBE) - A CUBE IS PAINTED ON ALL FACES AND THEN CUT INTO 1331 SMALLER CUBES. HOW MANY CUBES WILL HAVE AT LEAST ONE FACE PAINTED?
– ANSWER: 1049
MIXED DICE AND CUBE PUZZLES
- A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY OF GETTING A SUM OF 8?
ANSWER: 5/36 (POSSIBLE PAIRS: (2,6), (3,5), (4,4), (5,3), (6,2)) - A DIE IS THROWN THREE TIMES. WHAT IS THE PROBABILITY OF GETTING AT LEAST ONE EVEN NUMBER?
ANSWER: 7/8 - A CUBE IS PAINTED AND THEN CUT INTO 125 SMALLER CUBES. HOW MANY WILL HAVE EXACTLY TWO FACES PAINTED?
ANSWER: 48 - A CUBE IS PAINTED AND THEN CUT INTO 729 SMALLER CUBES. HOW MANY WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 665 - A DIE IS PLACED SO THAT THE NUMBERS 3, 4, AND 5 ARE VISIBLE. WHAT IS THE SUM OF THE HIDDEN NUMBERS?
ANSWER: 9 (TOTAL SUM = 21, VISIBLE SUM = 12, SO HIDDEN SUM = 9)
SECTION 13: ADVANCED DICE AND CUBE REASONING (CONTINUED)
DICE PROBABILITY AND LOGICAL QUESTIONS
- A DIE IS ROLLED ONCE. WHAT IS THE PROBABILITY OF GETTING A NUMBER GREATER THAN 2?
ANSWER: 4/6 = 2/3 (NUMBERS: 3, 4, 5, 6) - TWO DICE ARE ROLLED TOGETHER. WHAT IS THE PROBABILITY OF GETTING A SUM OF 11?
ANSWER: 2/36 = 1/18 (POSSIBLE PAIRS: (5,6), (6,5)) - IF A DIE IS ROLLED TWICE, WHAT IS THE PROBABILITY THAT THE SECOND ROLL IS EQUAL TO OR GREATER THAN THE FIRST?
ANSWER: 21/36 = 7/12 (COUNT ALL VALID PAIRS) - A DIE IS THROWN. WHAT IS THE PROBABILITY OF GETTING AN ODD NUMBER OR A NUMBER GREATER THAN 4?
ANSWER: 4/6 = 2/3 (NUMBERS: 1, 3, 5, 6) - A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY THAT NEITHER ROLL RESULTS IN A 4?
ANSWER: (5/6) × (5/6) = 25/36
DICE OPPOSITE FACES AND ARRANGEMENTS
- A STANDARD DIE IS PLACED SUCH THAT 3 IS ON TOP AND 1 IS IN FRONT. WHAT NUMBER IS OPPOSITE TO 3?
ANSWER: 4 (SINCE OPPOSITE FACES SUM TO 7) - A DIE HAS NUMBERS 2, 3, AND 5 ON ADJACENT FACES. WHAT NUMBER IS OPPOSITE TO 2?
ANSWER: 6 - A CUBE IS PLACED SUCH THAT 1, 2, AND 6 ARE VISIBLE. WHAT NUMBER IS OPPOSITE TO 6?
ANSWER: 5 - A DIE IS POSITIONED SO THAT THE SUM OF ADJACENT FACES IS ALWAYS EVEN. WHICH NUMBER MUST BE OPPOSITE TO 5?
ANSWER: 2 - A DIE IS POSITIONED SUCH THAT 4 IS AT THE TOP, 5 IS AT THE FRONT, AND 6 IS AT THE RIGHT. WHAT NUMBER IS AT THE BOTTOM?
ANSWER: 3
CUBE CUTTING AND PAINTING QUESTIONS
- A CUBE IS PAINTED ON ALL FACES AND CUT INTO 1000 SMALLER CUBES. HOW MANY CUBES HAVE THREE FACES PAINTED?
ANSWER: 8 (CORNER CUBES) - A CUBE OF 9×9×9 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY WILL HAVE EXACTLY TWO FACES PAINTED?
ANSWER: 96 (EDGE CUBES) - A CUBE OF 8×8×8 IS PAINTED AND THEN CUT INTO SMALLER CUBES. HOW MANY WILL HAVE EXACTLY ONE FACE PAINTED?
ANSWER: 192 (FACE-CENTER CUBES) - A CUBE IS DIVIDED INTO 1331 SMALLER CUBES. HOW MANY WILL HAVE NO PAINT AT ALL?
ANSWER: 729 (INNER 9×9×9 CUBE) - A CUBE IS PAINTED ON ALL FACES AND THEN CUT INTO 2197 SMALLER CUBES. HOW MANY CUBES WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 1729
MIXED DICE AND CUBE PUZZLES
- A DIE IS ROLLED TWICE. WHAT IS THE PROBABILITY OF GETTING A SUM OF 12?
ANSWER: 1/36 (POSSIBLE PAIR: (6,6)) - A DIE IS THROWN THREE TIMES. WHAT IS THE PROBABILITY OF GETTING AT LEAST ONE 4?
ANSWER: 91/216 (USE COMPLEMENT METHOD: 1 – (5/6 × 5/6 × 5/6)) - A CUBE IS PAINTED AND THEN CUT INTO 1331 SMALLER CUBES. HOW MANY WILL HAVE EXACTLY TWO FACES PAINTED?
ANSWER: 216 - A CUBE IS PAINTED AND THEN CUT INTO 3375 SMALLER CUBES. HOW MANY WILL HAVE AT LEAST ONE FACE PAINTED?
ANSWER: 2809 - A DIE IS PLACED SO THAT THE NUMBERS 1, 3, AND 5 ARE VISIBLE. WHAT IS THE SUM OF THE HIDDEN NUMBERS?
ANSWER: 12 (TOTAL SUM = 21, VISIBLE SUM = 9, SO HIDDEN SUM = 12)
HERE ARE SOME SPECIAL KEY PHRASES TAILORED FOR YOUR ARTICLE ON DICE AND CUBES – REASONING FOR PSU EXAMS TO IMPROVE SEO AND SEARCH VISIBILITY:
SPECIAL KEY PHRASES:
- HOW TO SOLVE DICE AND CUBE REASONING QUESTIONS EASILY
- DICE AND CUBE TRICKS FOR PSU AND COMPETITIVE EXAMS
- BEST SHORTCUT METHODS FOR DICE AND CUBE PROBLEMS
- MOST IMPORTANT DICE AND CUBE QUESTIONS FOR PSU EXAMS
- DICE AND CUBE APTITUDE QUESTIONS WITH SOLUTIONS
- UNDERSTANDING DICE OPPOSITE FACES AND ADJACENT FACES
- CUBE CUTTING AND PAINTING PROBLEMS EXPLAINED
- DICE AND CUBE LOGICAL REASONING FOR GOVERNMENT EXAMS
- STEP-BY-STEP GUIDE TO SOLVING CUBE-BASED PROBLEMS
- MASTER DICE PROBABILITY QUESTIONS FOR PSU EXAMS
- DICE AND CUBE MCQS WITH DETAILED EXPLANATIONS
- ADVANCED DICE AND CUBE REASONING TECHNIQUES
- COMMON MISTAKES TO AVOID IN DICE AND CUBE QUESTIONS
- HOW TO QUICKLY DETERMINE THE OPPOSITE FACE OF A DIE
- BEST DICE AND CUBE REASONING PRACTICE QUESTIONS
- DICE ROLLING PROBABILITY TRICKS FOR COMPETITIVE EXAMS
- NON-VERBAL REASONING: DICE AND CUBE PROBLEM-SOLVING
- ESSENTIAL FORMULAS FOR DICE AND CUBE APTITUDE QUESTIONS
- DICE AND CUBE PLACEMENT PUZZLES FOR SSC AND PSU EXAMS
- EFFECTIVE STRATEGIES FOR SOLVING CUBE ARRANGEMENT QUESTIONS
