Seven identical cubes each carry 20 dots. Eight at the corners and twelve at the edge midpoints. Each cube is cut once by a flat plane through some dots. Make the exposed face take each of the seven named shapes.
Your answer
Watch outAny shape seems possible, but with six faces the cut has at most six sides.
How to solve
StrategyVisualize Spatial Relationships — Guessing planes at random is hopeless; what steers the search is one rule — the number of sides of the cross-section equals the number of faces the plane cuts through. So I decide the number of sides first (3, 4 or 6), which tells me how many faces to aim at, and only then look for the particular plane. Two more rules do the rest: sides lying in opposite faces come out parallel, and equal-looking sides really are equal whenever the cube's own symmetry swaps them. Naming the eight vertices lets me write each answer as a short list of points instead of a vague description, and slicing a real eraser or a lump of clay is the check the book itself recommends.
1STEP 1
Name the corners so every cut can be written down
Give the corners names so cuts can be written down.
Naming points first is not busywork: without names it is impossible to say which dots you joined, and almost impossible to check whether the shape you drew is really the shape you wanted.
K.G.B.4Draw A Diagram
2STEP 2
The key rule: sides come from faces, one side per face
One side appears per face crossed.
number of sides = number of faces the plane cuts, 3 ≤ sides ≤ 6
This one sentence converts 'find a plane' into 'choose which faces to hit', and choosing three or four or six faces out of six is something you can do by pointing at a cube with your finger.
7.G.A.3Make A Systematic List
Each side of a cross-section comes from one face of the cube, so the number of sides can never beat the number of faces cut.
Why?
A flat cut meets a flat face in a straight line, and it can meet one face in only one such line.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The cross-section's outline is those straight pieces joined end to end with no gap, so the sides are exactly the faces met.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
3STEP 3
Shape 1 — equilateral triangle: slice a corner off
Slicing a corner gives an equilateral triangle.
△ BDG: BD = DG = GB = face diagonal
Every corner of a cube is surrounded by three identical square faces, so cutting the corner off symmetrically has to produce a triangle with three identical sides. You get the same equilateral triangle at any of the eight corners.
7.G.A.3Visualize Spatial Relationships
4STEP 4
Shape 2 — isosceles but not equilateral: slide one corner of that triangle down
Sliding one corner down gives an isosceles triangle.
Symmetry gives you 'isosceles' for free; the only thing left to check is that it is not accidentally equilateral, and one Pythagoras comparison settles that. Pulling one vertex of the corner cut down the edge is the natural way to break the equilateral triangle without breaking the symmetry.
8.G.B.7Visualize Spatial Relationships
5STEP 5
Shape 3 — square: cut straight across the middle
A straight middle cut gives a square.
each side = edge, each angle = 90°
Slicing a cube parallel to a face is like cutting a slice of bread from a loaf: the cut face is a copy of the end of the loaf.
7.G.A.3Draw A Diagram
6STEP 6
Shape 4 — rectangle that is not a square: cut through two opposite edges
Through two opposite edges gives a rectangle.
FG = AD = 2, GD = FA = 2√(2), 2 ≠ 2√(2)
Any plane through two opposite edges of a cube gives this same tall rectangle - it is the biggest flat rectangle you can cut out of a cube, and it is never a square because a diagonal of a square is always longer than its side.
8.G.B.7Visualize Spatial Relationships
7STEP 7
Shape 5 — trapezoid that is not a rectangle: one long parallel side and one short one
Unequal parallel sides give a trapezoid.
M_ABM_AD = 1/2BD = 1/2FH, √(2) ≠ 2√(2)
To get a trapezoid rather than a parallelogram you need the plane to hit two opposite faces in chords of different lengths - here the bottom face is crossed along its full diagonal while the top face is only clipped near corner A.
5.G.B.4Visualize Spatial Relationships
8STEP 8
Shape 6 — rhombus that is not a square: cut along a cube diagonal
Tilting the rectangle of shape 4 about its centre until its corners slide to edge midpoints turns four right angles into two sharp and two blunt ones while keeping all four sides equal - which is precisely a rhombus that is not a square.
5.G.B.4Visualize Spatial Relationships
9STEP 9
Shape 7 — regular hexagon: hit all six faces at once
Hitting all six faces gives a regular hexagon.
each side = 1/2×face diagonal = √(2) (edge 2)
This is the only way to get six sides, and it is worth doing with clay: cut a cube exactly through its centre, square to a long diagonal, and a perfect hexagon appears where you expected another square.
7.G.A.3Visualize Spatial Relationships
10STEP 10
Slice a real cube to check every one
Slicing for real shows all seven work.
Cross-sections are hard to trust from a drawing, because the oblique picture stretches every shape. A real cut removes all doubt, and the book suggests exactly this.
5.G.B.4Create A Physical Representation
Answer
all seven are possible
3, 4, 5, 6
Every one of the seven cuts uses only marked points — corners and edge midpoints — as required, and every side count is possible for a cube: 3, 3, 4, 4, 4, 4 and 6 faces cut, never more than the 6 faces available. The 'not' conditions are all checked by an actual comparison rather than by eye: cut 2 has √(5) ≠ 2√(2), so it is not equilateral; cut 4 has 2 ≠ 2√(2), so it is not a square; cut 5 has parallel sides √(2) and 2√(2), so it is not a parallelogram and therefore not a rectangle; cut 6 has unequal diagonals 2√(3) and 2√(2), so it is a rhombus and not a square. The side rule also tells you where the limits are: because a plane meets each flat face in at most one segment and a cube has only 6 faces, a cross-section of a cube can never have 7 or more sides — a heptagonal cross-section is impossible, and so is any shape needing two separate cuts of the same face.
Takeaway
Count faces, not corners: whichever faces your flat cut passes through become the sides of the cross-section, so a cube can give you three, four, five or six sides - and never more than six.
Name the corners so every cut can be written down
The key rule: sides come from faces, one side per face
Shape 1 — equilateral triangle: slice a corner off
Shape 2 — isosceles but not equilateral: slide one corner of that triangle down
Shape 3 — square: cut straight across the middle
Shape 4 — rectangle that is not a square: cut through two opposite edges
Shape 5 — trapezoid that is not a rectangle: one long parallel side and one short one
Shape 6 — rhombus that is not a square: cut along a cube diagonal
Shape 7 — regular hexagon: hit all six faces at once