Solid-Figure Problems: unit cubes, views, cube nets, drilled solids, open boxes, surface area, capacity, painted cubes, and cube motion.
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Questions 1–2 · 6 points each
The solid shown is built from congruent unit cubes, stacked from the base without gaps under the upper cubes. How many unit cubes are in the complete solid?
Original diagram, cropped to preserve the exact cube overlap.
Observe the three solids, each one unit cube high, made from congruent unit cubes. Use the same viewing directions for all three: front means looking straight at the broad front faces; right side means looking straight at the faces on the right; above means looking straight down. Compare only the outlines, not the internal dividing lines.
Viewing guide: front = square faces facing you; right side = slanted faces on the right; top = upper faces. Keep these directions fixed for all three solids.
Questions 3–6 · choose one answer · 6 points each
A cube of modeling clay has edge length 4 cm. It is cut into smaller cubes with edge length 2 cm. How many small cubes are obtained?
Which arrangement of six equal squares cannot be folded along its shared edges into a cube without overlap?
Candidate cube nets A–D
A cube has edge length 3 cm. Through the centers of the top and bottom, front and back, and left and right faces, a rectangular-prism hole with a square cross-section of side 1 cm is bored completely through the cube. What volume remains?
Original diagram, cropped to preserve the three tunnel openings and their perspective.
From a 13 cm by 9 cm rectangular cardboard sheet, a 2 cm square is cut from each corner. The remaining sheet is folded along the dashed lines to form an open rectangular container. Ignore cardboard thickness and extra material for seams. What is the container’s volume?
The shaded corner squares are removed before folding.
Questions 7–10 · 6 points each
Solid A (left, labeled 甲) is a complete 2 × 2 × 2 block of unit cubes. Solid B (right, labeled 乙) is made by removing its upper front-right corner cube. Compare their volumes and total surface areas, including the underside and all faces exposed by the removal. Fill each blank with greater than, equal to, or less than.
Solid A = left (甲). Solid B = right (乙). All small cubes have the same size.
The rectangular gift box is 60 cm long, 40 cm wide, and 30 cm high. The ribbon makes two perpendicular loops: one around the length and height, and one around the width and height. The knot uses an additional 20 cm of ribbon. How long is the complete ribbon?
Original diagram. The Chinese note states that 20 cm of ribbon is used for the knot.
The areas of the three different face types of a cuboid are 6, 8, and 12 square units. What is the cuboid’s volume?
The same capped bottle is shown upright and upside down; no water is lost. Its total height is 7 cm. The straight, wide part has a uniform cross-sectional area of 10 cm². Upright, the water is 4 cm deep, entirely within the straight part. Upside down, the water surface is 5 cm above the cap end, leaving the air in the straight part. Find the bottle’s capacity.
Original diagram of the upright and inverted bottle.
Questions 11–15 · show your reasoning · 12 points each
A cuboid’s length and width stay unchanged while its height increases by 5 dm, turning it into a cube. Its surface area increases by 160 dm². What was the original cuboid’s volume?
Every face of a cube with edge length 3 cm is painted green. The cube is then cut into unit cubes with edge length 1 cm. The newly cut faces are not painted. Among the resulting unit cubes, how many have:
Unit cubes with edge length 1 cm form a triangular staircase, with no gaps beneath the visible cubes. Each horizontal layer has rows that shorten by one cube: the bottom layer of a 5-layer solid has rows of 5, 4, 3, 2, and 1 cubes. The next layer has rows of 4, 3, 2, and 1, and so on up to one cube at the top, with the right-angle corners aligned. What is the total surface area of the 5-layer solid, including the underside?
Original diagram showing the first three stages of the layered pattern.
The diagram is a net of a cube. Numbers are placed on the six faces so that the numbers on each pair of opposite faces have a sum of 7. Find the values of A, B, and C.
Opposite face pairs must each total 7.
For this cube, the numbers on each pair of opposite faces have a sum of 7. Initially, 1 is on top, 2 is at the front facing you, and 3 is on the right. Each roll is a quarter-turn (90°) over an edge onto the next face. Keep your viewpoint fixed. Roll the cube backward (away from you) 15 times, then to your right 30 times. What number is on the top face at the end?
Starting position: top 1, front 2, right 3. ↑ Backward = away from you. → Right = your right. Keep your viewpoint fixed.
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