WORKED EXAMPLE. How many unit cubes make up this solid altogether?
(Total for Question 0 is 0 marks)
1
How many unit cubes make up this solid altogether?
(Total for Question 1 is 1 mark)
2
How many individual unit cubes are touching the table (the surface the solid stands on)?
(Total for Question 2 is 1 mark)
3
How many unit cubes make up this solid altogether?
(Total for Question 3 is 1 mark)
4
This solid stands inside an invisible box measuring 2 x 2 x 2 unit cubes (so a completely full box would need 8 cubes in total). How many extra unit cubes would need to be added to fill the box completely, with no gaps?
(Total for Question 4 is 1 mark)
5
How many unit cubes make up this solid altogether?
(Total for Question 5 is 1 mark)
6
How many individual unit cubes are touching the table (the surface the solid stands on)?
(Total for Question 6 is 1 mark)
7
This solid stands inside an invisible box measuring 3 x 2 x 3 unit cubes (so a completely full box would need 18 cubes in total). How many extra unit cubes would need to be added to fill the box completely, with no gaps?
(Total for Question 7 is 1 mark)
8
A unit cube counts as completely hidden if it has no visible left, right, front, back or top face - the underside resting on the table does not count. How many unit cubes in this solid are completely hidden from view?
(Total for Question 8 is 1 mark)
9
How many unit cubes make up this solid altogether?
(Total for Question 9 is 1 mark)
10
The solid is lifted, dipped completely in paint so that every outward-facing surface (including the underside) is coloured, then rebuilt in exactly the same arrangement. How many individual unit cubes have exactly 2 painted faces?
(Total for Question 10 is 1 mark)
11
How many unit cubes make up this solid altogether?
(Total for Question 11 is 1 mark)
12
How many individual unit cubes are touching the table (the surface the solid stands on)?
(Total for Question 12 is 1 mark)
13
A unit cube counts as completely hidden if it has no visible left, right, front, back or top face - the underside resting on the table does not count. How many unit cubes in this solid are completely hidden from view?
(Total for Question 13 is 1 mark)
14
A unit cube counts as completely hidden if it has no visible left, right, front, back or top face - the underside resting on the table does not count. How many unit cubes in this solid are completely hidden from view?
(Total for Question 14 is 1 mark)
15
This solid stands inside an invisible box measuring 3 x 3 x 3 unit cubes (so a completely full box would need 27 cubes in total). How many extra unit cubes would need to be added to fill the box completely, with no gaps?
(Total for Question 15 is 1 mark)
16
A unit cube counts as completely hidden if it has no visible left, right, front, back or top face - the underside resting on the table does not count. How many unit cubes in this solid are completely hidden from view?
(Total for Question 16 is 1 mark)
17
How many unit cubes make up this solid altogether?
(Total for Question 17 is 1 mark)
18
The solid is lifted, dipped completely in paint so that every outward-facing surface (including the underside) is coloured, then rebuilt in exactly the same arrangement. How many individual unit cubes have exactly 3 painted faces?
(Total for Question 18 is 1 mark)
19
The solid is lifted, dipped completely in paint so that every outward-facing surface (including the underside) is coloured, then rebuilt in exactly the same arrangement. How many individual unit cubes have exactly 4 painted faces?
(Total for Question 19 is 1 mark)
20
The solid is lifted and every outward-facing surface, including the underside, is painted red. How many unit squares (faces) are painted red in total?
(Total for Question 20 is 1 mark)
21
This solid stands inside an invisible box measuring 3 x 3 x 4 unit cubes (so a completely full box would need 36 cubes in total). How many extra unit cubes would need to be added to fill the box completely, with no gaps?
(Total for Question 21 is 1 mark)
22
This solid has a tall centre column. Answer both parts about the same solid.
(a)How many unit cubes make up this solid altogether?(1)
A) 27
B) 28
C) 29
D) 30
E) 31
(b)A unit cube counts as completely hidden if it has no visible left, right, front, back or top face - the underside resting on the table does not count. How many unit cubes in this solid are completely hidden from view?(1)
A) 1
B) 2
C) 3
D) 4
E) 5
(Total for Question 22 is 2 marks)
Mark scheme · EP.N10 3D Block Counting
Question 0
B0 Worked example - no marks awarded. Demonstrates the method: add up the height of every stack in the base grid, one grid square at a time, including squares of height 1.
Answer: C
Question 1
B1 Correct option identified (6 oe) - the four stack heights 1, 2, 2 and 1 sum to 6
Answer: 6
Question 2
B1 Correct option identified (3 oe) - a cube touches the table wherever the grid square is occupied (height at least 1); three of the four grid squares are occupied
Answer: 3
Question 3
B1 Correct option identified (8 oe) - the six stack heights 1, 1, 2, 1, 1 and 2 sum to 8
Answer: 8
Question 4
B1 Correct option identified (3 oe) - the solid uses 5 cubes, so 8 minus 5 leaves 3 cubes needed to complete the box
Answer: 3
Question 5
B1 Correct option identified (10 oe) - the six stack heights sum to 10
Answer: 10
Question 6
B1 Correct option identified (6 oe) - six of the nine grid squares are occupied (A1, C1, A2, B2, B3, C3)
Answer: 6
Question 7
B1 Correct option identified (8 oe) - the solid uses 10 cubes, so 18 minus 10 leaves 8 cubes needed to complete the box
Answer: 8
Question 8
B1 Correct option identified (2 oe) - only the bottom two cubes of the centre (B2) stack are surrounded on all four sides and covered from above
Answer: 2
Question 9
B1 Correct option identified (18 oe) - each row of stack heights sums to 6, and 6 + 6 + 6 = 18
Answer: 18
Question 10
B1 Correct option identified (1 oe) - only the single centre cube (B2) has all four side faces covered by neighbours, leaving just its top and bottom painted
Answer: 1
Question 11
B1 Correct option identified (12 oe) - each row of stack heights sums to 6, and 6 + 6 = 12
Answer: 12
Question 12
B1 Correct option identified (5 oe) - five of the eight grid squares are occupied (A1, C1, D1, B2, C2)
Answer: 5
Question 13
B1 Correct option identified (3 oe) - the bottom three cubes of the centre (B2) stack are each surrounded on all four sides and covered from above
Answer: 3
Question 14
B1 Correct option identified (2 oe) - the shortest neighbour of the centre stack limits full cover to only the bottom two levels
Answer: 2
Question 15
B1 Correct option identified (7 oe) - the solid uses 20 cubes, so 27 minus 20 leaves 7 cubes needed to complete the box
Answer: 7
Question 16
B1 Correct option identified (4 oe) - the bottom two levels of both interior stacks (B2 and C2) are hidden, giving 2 + 2 = 4
Answer: 4
Question 17
B1 Correct option identified (23 oe) - the three row totals 8, 7 and 8 sum to 23
Answer: 23
Question 18
B1 Correct option identified (3 oe) - the three bottom cubes of the height-2 stacks that sit at the front or back edge (A1, A2 and A3) each have exactly 3 painted faces
Answer: 3
Question 19
B1 Correct option identified (5 oe) - the four corner cubes of the flat layer plus the middle cube of the centre tower each end up with exactly 4 painted faces
Answer: 5
Question 20
B1 Correct option identified (20 oe) - adding the exposed faces of all five cubes gives 4 + 4 + 4 + 3 + 5 = 20
Answer: 20
Question 21
B1 Correct option identified (9 oe) - the solid uses 27 cubes, so 36 minus 27 leaves 9 cubes needed to complete the box
Answer: 9
Question 22
(a) B1 Correct option identified (29 oe) - the three row totals 7, 15 and 7 sum to 29
(a) Answer: 29
(b) B1 Correct option identified (3 oe) - the bottom three cubes of the centre (B2) stack are each surrounded on all four sides and covered from above