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Question 1 / 20
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1
Identify the box that is formed when the provided paper sheet (X) is folded into a cube.
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Solution: Step 1: Recognize the given sheet (X) as a cube net, similar to Form III. Step 2: Determine the pairs of opposite faces: The half-shaded face appears opposite the face bearing a rhombus. The face with a black circle appears opposite one of the two blank faces. The face with a '+' sign appears opposite the other blank face. Step 3: Analyze figures (1) and (4): Both show the half-shaded face adjacent to the rhombus, which is incorrect as they must be opposite. So, (1) and (4) cannot be formed. Step 4: Analyze figure (3): While its faces (black circle, '+' sign, blank) could theoretically be adjacent, the specific orientation and arrangement shown in figure (3) cannot be achieved by folding and rotating the net (X). Step 5: Conclude that only the cube shown in figure (2) can be formed, as its face arrangement and relative orientations are consistent with the net.
2
From the four positions of a dice given below, find the color which is opposite to yellow?
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Solution: Step 1: (Assume four images of a dice showing colors are provided). Step 2: Identify all colors that are adjacent to 'yellow' from all given positions of the dice. Step 3: The solution states: "The colours adjacent to yellow are orange, blue, red and rose." Step 4: A dice has 6 faces. If 'yellow' is on one face, and four other colors (orange, blue, red, rose) are adjacent to it, then these five colors cannot be opposite to yellow. Step 5: By elimination, the only remaining color (from a typical set of 6 colors used in such problems, or from the options provided) must be opposite to yellow. The remaining color is 'Violet'. Step 6: Conclude that 'Violet' is the color opposite to yellow.
3
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.
4
Identify the color of the face opposite to the brown face.
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Solution: Step 1: (This problem requires visual input or a descriptive context about the cube's orientation and coloring, which is not provided in the input JSON. Assuming such context is available to the solver). Step 2: Based on the implied visual or textual context, determine the face that lies directly opposite the brown face. Step 3: The face opposite to brown is White.
5
From the provided images illustrating two positions of a dice, ascertain the number of points on the face opposite to the face that contains 2 points.
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Solution: Step 1: (Assume two images of a dice are provided, showing points). Step 2: Identify common faces and their positions. The solution states that '5 points' is a common face and is in the same position in the first two positions of the dice. Step 3: Apply the dice rule: If two positions of a dice show one common face in the same position, then the remaining visible faces will be opposite to each other in their corresponding positions. Step 4: Based on this rule, by comparing the other visible faces when '5' is common and in the same position, the face containing '2 points' will be opposite to the face containing '6 points'. Step 5: Conclude that '6 points' will be on the face opposite to the face which contains '2 points'.
6
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.
7
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.
8
A dice marked with alphabets A-F is rolled thrice, showing three positions. Find the alphabet opposite A.
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Solution: Step 1: Compare figures (ii) and (iii). The common face between them is 'E'. Step 2: List all alphabets adjacent to 'E' from both figures: C (from ii), D (from ii), B (from iii), and F (from iii). Step 3: A dice has 6 faces with alphabets A-F. The alphabets B, C, D, F are adjacent to E. Step 4: The only remaining alphabet that has not been listed as adjacent to E is A. Step 5: Therefore, the alphabet A appears opposite E. Conversely, E appears opposite A.
9
Observe the two provided positions of a single dice. Determine the number of points on the face opposite to the face showing 5 points.
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Solution: Step 1: (Assume two images of a dice are provided, showing points). Step 2: Identify the common face(s) and their positions in both dice views. The solution indicates that '1 point' is a common face and is in the same position in both views. Step 3: Apply the dice rule: If two different positions of a dice are shown, and one common face is in the same position, then the remaining faces visible in corresponding positions will be opposite to each other. Step 4: Based on this rule and the problem's configuration (assuming a standard setup where '5' and '4' would be in corresponding positions relative to the common '1'), the face opposite to '5 points' is '4 points'. Step 5: Conclude that '4 points' will appear on the face opposite to the face containing '5 points'.
10
Given four positions of a dice, determine the number on the bottom face when the dice is in position (iii).
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Solution: Step 1: In Figure (iii), the top face is 3. We need to find the number opposite 3, which will be at the bottom face. 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 4, 5, and 6 are visible. This indicates 5 and 6 are adjacent to 4. Step 4: Examine Figure (ii), where 1, 4, and 2 are visible. This indicates 1 and 2 are adjacent to 4. Step 5: Examine Figure (iv), where 5, 2, and 4 are visible. This indicates 5 and 2 are adjacent to 4. Step 6: Combining observations from Figures (i), (ii), and (iv), the numbers 1, 2, 5, and 6 are all adjacent to 4. Step 7: On a six-faced dice, if 1, 2, 5, 6 are adjacent to 4, then the only remaining number, 3, must be opposite to 4. Step 8: Therefore, 3 is opposite 4. Step 9: Since 3 is at the top in Figure (iii), the number at the bottom face must be 4.
11
Determine the total count of small cubes that will have only one face colored.
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Solution: Step 1: Small cubes with only one colored face are located on the central part of each face of the larger cuboid, excluding edges and corners. Step 2: The formula for calculating such cubes is 2 * [(L-2) x (B-2) + (B-2) x (H-2) + (H-2) x (L-2)], where L, B, and H are the dimensions of the larger cuboid. Step 3: Based on the calculation (4-2) x (3-2) for no-face-colored cubes (from question 12673), the implied dimensions for the cuboid in this context are 4x3x3. Step 4: Apply the formula with these dimensions: 2 * [ (4-2) x (3-2) + (3-2) x (3-2) + (3-2) x (4-2) ] = 2 * [ (2 x 1) + (1 x 1) + (1 x 2) ] = 2 * [ 2 + 1 + 2 ] = 2 * 5 = 10. Step 5: Therefore, 10 small cubes will have only one face colored.
12
Determine the number of cubes that each have two adjacent blue faces.
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Solution: Step 1: Refer to the larger problem context which describes different sets or types of cubes and their coloring. Step 2: From the context, identify the specific group of cubes where each cube has two adjacent blue faces. Step 3: The solution explicitly states that 'Second 64 cubes are such each of whose two faces are blue'. Step 4: Therefore, 64 cubes have two adjacent blue faces.
13
If all small cubes having both black and green colors are removed, how many cubes will be left?
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Solution: Step 1: Identify the total number of small cubes, which is 24. Step 2: Identify the specific group of small cubes that are colored both black and green. Based on the implied problem conditions, this number is given as 8. Step 3: To find the number of remaining cubes, subtract the count of cubes removed from the total count. Step 4: Perform the calculation: 24 - 8 = 16. Step 5: Therefore, 16 cubes will remain after removing those with both black and green colors.
14
Given two positions of a dice, how many points will be on the top when 2 points are at the bottom?
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Solution: Step 1: (Assume two images of a dice showing points are provided, e.g., View A: (Top:3, Front:2, Right:5) and View B: (Top:3, Front:1, Right:4)). Step 2: Identify the common face and its position in both views. From the assumed views, '3 points' is common and is in the same relative position (e.g., both are on top). Step 3: Apply the dice rule: When a common face is in the same position in two different views of a dice, the other faces in corresponding positions are opposite to each other. Step 4: Compare the faces: In View A, '2 points' is in the front position. In View B, '1 point' is in the front position. Step 5: Therefore, '2 points' is opposite to '1 point'. Step 6: Conclude that if 2 points are at the bottom, then 1 point will be on the top.
15
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.
16
Determine the count of small cubes that possess exactly four colored faces and two uncolored faces.
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Solution: Step 1: Understand the specific condition: '4 coloured sides and two non-coloured sides'. For standard cube cutting and coloring, a small cube can have a maximum of 3 colored faces (at the corners of the larger block). Step 2: This condition implies a very specific or non-standard coloring pattern or cube arrangement, likely detailed in an accompanying diagram not provided here. Step 3: Based on the provided solution, which states 'Only 4 cubes situated at the corners of the cuboid will have 4 coloured and 2 non-coloured sides', we infer that these particular 4 corner cubes of the cuboid (likely the 6x4x1 block mentioned previously) uniquely meet this criterion under the problem's specific rules. Step 4: Therefore, 4 cubes will have 4 colored sides and two non-colored sides.
17
From the displayed positions of a cube, determine which letter will be found on the face directly opposite to the face marked with 'A'.
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Solution: Step 1: Observe the given positions of the cube and the letters visible on its faces. Step 2: Identify all letters that are shown on faces adjacent to 'A' in any of the provided views. Step 3: The explanation states that 'The letters of the adjacent faces to the face with letter A, are B, F, C and E.' Step 4: Since a standard cube has six faces, and four faces (B, F, C, E) are adjacent to 'A', the one remaining face must be opposite to 'A'. Step 5: Therefore, the letter 'D' is on the face opposite to 'A'.
18
Determine the count of small cubes where one face is green and another is either black or red.
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Solution: Step 1: Understand the coloring conditions: one green face and a second face that can be either black or red. Step 2: Based on the implied cube structure and coloring rules (from a larger problem context), identify the number of edge cubes that satisfy one of the conditions (e.g., green and black). The solution implies this number is 8 for each condition. Step 3: Since the condition is 'either black or red', multiply this base count by the number of possibilities: 8 * 2 = 16. Step 4: The total number of small cubes satisfying the condition is 16.
19
Given two positions of a dice, determine the number that will be at the top when 2 is at the bottom.
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Solution: Step 1: Observe the two given positions of the dice. Step 2: Identify the common number visible in both positions, which is '3'. Step 3: From the common number (3), trace the visible faces in a clockwise direction for both figures. Step 4: In Figure (i), starting from 3 and moving clockwise: 3 -> 5 -> 2. Step 5: In Figure (ii), starting from 3 and moving clockwise: 3 -> 6 -> 1. Step 6: By comparing the sequences, the numbers at corresponding positions are opposite to each other. Thus, 5 is opposite 6, and 2 is opposite 1. Step 7: The remaining number, 4, must be opposite the common face, 3. Step 8: So, the opposite pairs are (5 vs 6), (2 vs 1), and (3 vs 4). Step 9: The question asks for the number at the top when 2 is at the bottom. Since 2 is opposite 1, the number at the top will be 1.
20
How many small cubes will have precisely two colored faces?
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Solution: Step 1: Small cubes with exactly two colored faces are typically those located on the edges of the larger cuboid but not at its corners. Step 2: The provided solution states the count as a summation: '6 from the front + 6 from the back + 2 from the left + 2 from the right'. Step 3: Sum these individual counts: 6 + 6 + 2 + 2 = 16. Step 4: Therefore, 16 small cubes will have only two faces colored, according to this specific counting method.
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