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Question 1 / 20
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1
Given two distinct positions of a single die, if the face with 3 points is oriented downwards, what number of points will be visible on the top face?
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Solution: Step 1: Analyze the two provided positions of the dice to identify common faces or deduce relationships between faces. Step 2: Apply the rules for finding opposite faces on a die (e.g., if one number is common in two positions, rotate clockwise/anti-clockwise from that common face to find opposite pairs; if two numbers are common, the remaining non-common numbers are opposite). Step 3: The solution states that 'According to the rule (2) when 3 points are at the bottom then 4 points will be at the top.' This implies that the faces with 3 points and 4 points are opposite to each other. Step 4: Therefore, if 3 points are at the bottom, the top face will have 4 points.
2
Given four different positions of a dice, determine the number opposite 3.
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Solution: Step 1: The goal is to find the number opposite 3. Step 2: To find the opposite of 3, we can identify all numbers adjacent to it from the given figures. Step 3: Examine Figure (i), where 2, 3, and 6 are visible. This indicates 2 and 6 are adjacent to 3. Step 4: Examine Figure (iii), where 3, 1, and 5 are visible. This indicates 1 and 5 are adjacent to 3. Step 5: Examine Figure (iv), where 3, 5, and 1 are visible. This confirms 1 and 5 are adjacent to 3. Step 6: Combining these observations, the numbers 1, 2, 5, and 6 are all adjacent to 3. Step 7: On a six-faced dice, if 1, 2, 5, 6 are adjacent to 3, then the only remaining number, 4, must be opposite to 3. Step 8: Therefore, the number opposite 3 is 4.
3
How many small cubes will possess exactly three colored faces?
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Solution: Step 1: Small cubes with three colored faces are always located at the corners of the larger cuboid after it has been painted and cut. Step 2: A standard three-dimensional cuboid has 8 corners. Step 3: Therefore, 8 small cubes will have three faces colored.
4
Select all boxes that can be formed from the given paper sheet (X) by folding.
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Solution: Step 1: Recognize the given sheet (X) as a cube net, similar to Form II. Step 2: Determine the pairs of opposite faces: The two half-shaded faces lie opposite each other. One of the three blank faces appears opposite to the face bearing a dot. The remaining two blank faces are opposite each other. Step 3: Examine each cube option (1, 2, 3, 4). Step 4: Verify that no option shows two half-shaded faces adjacent to each other. Step 5: Verify that the dot face is never shown adjacent to the blank face it's opposite. Step 6: Since all four cubes (1), (2), (3), and (4) demonstrate face arrangements consistent with the derived opposite pairs and can be achieved through valid rotations, all of them can be formed by folding sheet (X).
5
Calculate the number of small cubes that remain uncolored on all faces.
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Solution: Step 1: Identify the dimensions of the larger cuboid from the context implied by the solution: Length = 5, Width = 4, Height = 3. Step 2: Apply the formula for the number of small cubes with no faces colored in a cuboid: (Length - 2) * (Width - 2) * (Height - 2). Step 3: Substitute the dimensions into the formula: (5 - 2) * (4 - 2) * (3 - 2). Step 4: Perform the calculation: 3 * 2 * 1 = 6. Step 5: The total number of small cubes with no faces colored is 6.
6
Two views of a die, numbered with 1 to 6 dots on its faces, are presented. If the die is currently resting on the face with three dots, what will be the number of dots on the face at the top?
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Solution: Step 1: Examine the provided figures (i) and (ii) and the solution's deduction. Step 2: From figures (i) and (ii), the numbers 2, 4, and 6 are observed to be adjacent to 3 dots. (This implies these are the faces seen with 3 in some orientation, and 5 is not seen with 3). Step 3: A standard die has numbers 1 through 6. If 2, 4, and 6 are adjacent to 3, then the remaining numbers are 1 and 5. Step 4: One of these two remaining numbers (1 or 5) must be the fourth face adjacent to 3, and the other must be the face directly opposite to 3. Step 5: Since there isn't enough information to definitively distinguish between 1 and 5 as the opposite face to 3, both are possibilities. Step 6: If the die is resting on the side with three dots (meaning 3 is at the bottom), then the number of dots on the side at the top could be either 1 or 5.
7
Determine which of the given boxes can be formed by folding the sheet of paper (X).
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Solution: Step 1: Recognize the given sheet (X) as a cube net, similar to Form I. Step 2: Identify the pairs of opposite faces: one half-shaded face is opposite a blank face, the other half-shaded face is opposite another blank face, and the two remaining blank faces are opposite each other. Step 3: Analyze cubes (1) and (4): Both show faces that are consistent with the opposite face rules and relative orientations from the net, so they can be formed. Step 4: Analyze cubes (2) and (3): While their faces appear adjacent, the specific arrangement and orientation required to form these cubes cannot be achieved by folding the given sheet (X) without violating the inherent structure of the net. Step 5: Conclude that only cubes in figures (1) and (4) can be formed.
8
From the provided dice positions, determine the number of dots on the face opposite the one with three dots.
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Solution: Step 1: The goal is to find the number of dots opposite 3 dots. Step 2: To find the opposite of 3 dots, we can identify all faces adjacent to it from the given figures. Step 3: Examine Figure (i), where 3, 6, and 4 dots are visible. This indicates 6 dots and 4 dots are adjacent to 3 dots. Step 4: Examine Figure (ii), where 3, 1, and 2 dots are visible. This indicates 1 dot and 2 dots are adjacent to 3 dots. Step 5: Examine Figure (iv), where 3, 4, and 1 dots are visible. This confirms 4 dots and 1 dot are adjacent to 3 dots. Step 6: Combining these observations, the faces with 1, 2, 4, and 6 dots are all adjacent to the face with 3 dots. Step 7: On a six-faced dice, if faces with 1, 2, 4, 6 dots are adjacent to the face with 3 dots, then the only remaining number of dots, 5, must be opposite to 3 dots. Step 8: Therefore, there are 5 dots on the face opposite the one with three dots.
9
Count the number of small cubes that have only one face colored.
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Solution: Step 1: Infer that the problem refers to a larger cube divided into 4x4 small cubes on each face, making it a 4x4x4 cube. Step 2: For each face of the large cube, the number of small cubes with only one colored face is calculated as (N-2)^2, where N is the number of divisions along one edge. For N=4, this is (4-2)^2 = 2^2 = 4. Step 3: Since a cube has 6 faces, multiply the count per face by 6: 4 * 6 = 24. Step 4: The total number of small cubes with only one face colored is 24.
10
Identify the two colors that are present on an equal number of faces across the given cubes.
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Solution: Step 1: From the context, observe that the 'First 64 cubes' are described as having two green faces each. Step 2: From the context, observe that the 'second 64 cubes' are described as having two blue faces each. Step 3: Since both green and blue are each associated with 64 cubes (implying an equal total number of faces or equivalent representation in the overall structure), these two colors have the same 'number of faces' in this context. Step 4: Therefore, Blue and Green are the two colors that have the same number of faces.
11
Determine the count of small cubes that have no faces colored.
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Solution: Step 1: Identify the side length 'x' of the large cube from the solution, where x = 4. Step 2: Apply the formula for small cubes with no faces colored: (x - 2)^3. Step 3: Substitute the value of x into the formula: (4 - 2)^3. Step 4: Calculate the result: 2^3 = 8. Step 5: The total number of small cubes with no faces colored is 8.
12
Which of the provided boxes is identical to the cube formed by folding the given sheet of paper (X)?
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Solution: Step 1: Recognize the given sheet (X) as a cube net, similar to Form II. Step 2: Determine the pairs of opposite faces: The two half-shaded faces lie opposite each other. The face bearing a square lies opposite one of the two blank faces. The face bearing a circle lies opposite the other blank face. Step 3: Analyze figures (2) and (3): Both show the two half-shaded faces adjacent to each other, which is incorrect as they must be opposite. So, (2) and (3) cannot be formed. Step 4: Analyze figure (4): While its faces (square, circle, blank) could theoretically be adjacent, the specific orientation and arrangement shown in figure (4) cannot be achieved by folding and rotating the net (X). Step 5: Conclude that only the cube in figure (1) can be formed, as its face arrangement and relative orientations are consistent with the net.
13
Calculate the number of small cubes formed that have no faces colored.
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Solution: Step 1: Identify the side length 'x' of the large cube from the solution, where x = 5. Step 2: Apply the formula for small cubes with no faces colored: (x - 2)^3. Step 3: Substitute the value of x into the formula: (5 - 2)^3. Step 4: Calculate the result: 3^3 = 27. Step 5: The total number of small cubes with no faces colored is 27.
14
From the two provided positions of a single dice, numbered 1 through 6, determine the number opposite 3.
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Solution: Step 1: Observe the two given positions of the dice. Step 2: Identify the visible faces in Figure (i): 5, 2, 6. Step 3: Identify the visible faces in Figure (ii): 5, 1, 4. Step 4: From Figure (i), the numbers adjacent to 5 are 2 and 6. Step 5: From Figure (ii), the numbers adjacent to 5 are 1 and 4. Step 6: Combining these observations, the numbers 1, 2, 4, and 6 are all adjacent to 5. Step 7: A standard dice has 6 faces with numbers 1 through 6. If 1, 2, 4, 6 are adjacent to 5, then the only remaining number, 3, must be opposite to 5. Step 8: Therefore, the number opposite 3 is 5.
15
How many small cubes will have red, green, and black colors on at least one face?
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Solution: Step 1: Understand that small cubes with three distinct colored faces (Red, Green, Black) are typically found at the corners of the original larger cuboid after it has been painted and cut. Step 2: While the specific dimensions for this question are not explicitly given, the solution implies a configuration where 4 cubes meet this criterion. Step 3: If the original cuboid is a 6x4x1 flat slab, it has 4 corners. Step 4: Assuming a coloring scheme where these three distinct colors meet at these specific corner positions, then 4 such small cubes will be formed. Step 5: Therefore, 4 cubes will have red, green, and black colors on at least one side.
16
The image displays four distinct positions of a die. Determine the number on the face directly opposite to the face showing 6.
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Solution: Step 1: Compare figures (i) and (iii). Both figures show the number '5' as a common face. Step 2: In figure (i), moving clockwise from 5, the numbers are 4, then 3. Step 3: In figure (iii), moving clockwise from 5, the numbers are 6, then 2. Step 4: According to the common face rule, if a common face is in the same relative position (or if we trace clockwise), the numbers in corresponding positions are opposite to each other. Thus, 4 is opposite 6, and 3 is opposite 2. Step 5: The remaining numbers are 1 and 5, so 1 is opposite 5. Step 6: Based on this consistent set of opposite pairs ((1-5), (2-3), (4-6)), the number on the face opposite 6 is 4.
17
When the provided figure is folded into a cube, what number of dots will appear on the face opposite to the face with three dots?
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Solution: Step 1: Identify the arrangement of faces in the given unfolded figure (net), which is similar to Form V. Step 2: Apply the rule for identifying opposite faces in a cube net: Faces separated by one face in a straight line are opposite to each other. Step 3: Locate the face bearing three dots in the net. Step 4: Following the rule, the face bearing six dots will lie directly opposite the face bearing three dots. Step 5: Therefore, 6 dots lie opposite the face with three dots.
18
Calculate the total number of small cubes that have at most two faces colored.
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Solution: Step 1: Identify the number of small cubes with exactly two faces colored, which is given as 24. Step 2: Identify the number of small cubes with exactly one face colored, which is given as 24. Step 3: Identify the number of small cubes with no faces colored, which is given as 8. Step 4: 'At most two faces colored' implies summing the counts of cubes with 0, 1, or 2 colored faces. Step 5: Add these individual counts: 24 (two faces) + 24 (one face) + 8 (no face) = 56. Step 6: The total number of small cubes with at most two faces colored is 56.
19
Four standard dice are rolled, and their top faces show the numbers 4, 3, 1, and 5, resulting in a total sum of 13 for these top faces. What is the combined total of the numbers on the faces touching the ground?
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Solution: Step 1: Recall the fundamental property of a standard die: the sum of the numbers on any two opposite faces is always 7. Step 2: The numbers on the top faces are 4, 3, 1, and 5. Step 3: Determine the numbers on the faces touching the ground (opposite the top faces): * Opposite 4 is 7 - 4 = 3. * Opposite 3 is 7 - 3 = 4. * Opposite 1 is 7 - 1 = 6. * Opposite 5 is 7 - 5 = 2. Step 4: Calculate the total sum of these bottom faces: 3 + 4 + 6 + 2 = 15.
20
Given three distinct views (X, Y, and Z) of a single dice, determine the number that lies opposite 6.
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Solution: Step 1: Analyze positions X and Y. Step 2: Identify the common number visible in both positions X and Y, which is '4'. Step 3: Trace the visible faces in a clockwise direction starting from the common number. Step 4: In Position X (from 4 clockwise): 4 -> 1 -> 5. Step 5: In Position Y (from 4 clockwise): 4 -> 6 -> 3. Step 6: By comparing these sequences, the numbers at corresponding positions are opposite to each other. Therefore, 1 is opposite 6, and 5 is opposite 3. Step 7: The remaining number, 2, must be opposite the common face, 4. Step 8: So, the complete set of opposite pairs is (1 vs 6), (5 vs 3), and (2 vs 4). Step 9: The question asks for the number that lies opposite 6. Step 10: From our established pairs, the number opposite 6 is 1.
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