Problem
Count distinct stackings up to symmetry
Eight unit cubes build a solid three layers high. Seen from above it is a plus of five squares. Every cell carries at least one cube. Count the different stackings.
Reasoning · Grade 6-2 Counting Cubes and Stackings
Your answer
Watch out Different positions look like new stackings, but a turn can match them.
How to solve
Strategy Make a Systematic List — The two conditions turn out to fix the five heights completely -- they can only be 3, 2, 1, 1, 1 -- so the whole question becomes where to put the 3 and where to put the 2 on a plus. That is a small, finite listing job, and the safe way to do it without missing a case or writing one twice is to fix a viewing position first: turn the picture so the tall stack lands in a standard place, then list what is left. Handling real blocks settles the one delicate point, which is which pairs a turn can actually match up.
1STEP 1
Give every cell its first cube
One per cell leaves 3 cubes over.
8 - 5 = 3
Handing out one cube to every cell first is the same move as giving everybody one sweet before sharing out the rest, and it turns an 8-cube problem into a much smaller 3-cube problem.
2.OA.A.1Draw A Diagram2STEP 2
Work out the five heights
Three layers forces heights 3, 2, 1, 1, 1.
3 + 2 + 1 + 1 + 1 = 8
Once the spare cubes are down to three, you can check the handful of ways to split them by hand, and only one split survives both conditions.
2.OA.A.1Make A Systematic List3STEP 3
Choose where the 3 goes, using a turn to standardise the picture
Turn the picture so the 3 sits in one fixed place.
Turning the picture until the tall stack is in a fixed place is what stops the same solid being written down twice in two different orientations.
8.G.A.1Make A Systematic ListTurning the picture until the tall stack sits in a fixed place stops the same solid being written down twice.
Why?
A turn lays the solid onto a copy of itself, so a turned arrangement is the same solid seen from another side.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
Grouping the arrangements into families a turn can move between puts each in exactly one family, so counting families counts the different solids.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
4STEP 4
Case A: the 3 is on the top arm
With the 3 on an arm, the 2 has four places.
Once one cell is nailed down by a turn, the remaining choices are just four different places for a single cube, which is small enough to list on one line.
8.G.A.1Make A Systematic List5STEP 5
Check that the left-hand and right-hand versions really are different
Left and right versions stay different.
A mirror image is the one kind of look-alike that turning can never produce, which is exactly why this pair has to be checked with the blocks rather than assumed away.
8.G.A.1Create A Physical Representation6STEP 6
Case B: the 3 is on the centre
With the 3 in the centre there is one way.
When the tall stack sits at the very middle, spinning the solid a quarter turn at a time visits all four arms, so the four ways of placing the 2 are one solid looked at from four sides.
8.G.A.1Visualize Spatial Relationships7STEP 7
Collect the answers and check each one
Altogether that is 5.
3 + 2 + 1 + 1 + 1 = 8
Re-adding each finished picture is a cheap last guard against a stacking that looks right but quietly uses seven or nine cubes.
5.MD.C.4Make A Systematic ListAnswer
5 stackings
3 + 2 + 1 + 1 + 1 = 8
Every listed stacking uses 8 cubes on 5 cells with a tallest stack of exactly 3, so all five obey both conditions, and the count 5 matches the five blank grids the book prints. A separate count confirms the total: ignoring turns altogether there are 5 places for the 3 and then 4 places for the 2, which is 5 x 4 = 20 labelled pictures. Case B accounts for 4 of those 20 and collapses to 1, while Case A accounts for the other 16 and collapses to 4, because each of its four answers can be turned into four different pictures. That is 4 + 16 = 20 pictures folding into 1 + 4 = 5 solids, so nothing has been lost and nothing double-counted.
Takeaway
The conditions force the heights to be 3, 2, 1, 1, 1 -- then just turn the picture so the tall stack is at the top and count the places the 2 can go.
- Give every cell its first cube
- Work out the five heights
- Choose where the 3 goes, using a turn to standardise the picture
- Case A: the 3 is on the top arm
- Check that the left-hand and right-hand versions really are different
- Case B: the 3 is on the centre
- Collect the answers and check each one