Problem
Largest and smallest surface area
Twelve unit cubes are given. All twelve build a rectangular box with no gaps or overhang. Several different boxes are possible. Find the smallest possible surface area.
Reasoning · Grade 6-2 Surface Area of Cube Stacks
Your answer
Watch out Same cube count suggests same surface, but it depends on the shape.
How to solve
Strategy Make a Systematic List — The word 'smallest' means I have to be sure that nothing beats my answer, and the only honest way to be sure is to list every box that can be built and compare them all. Luckily there are very few: the three edge lengths are whole numbers whose product is 12, so writing the factor triples of 12 in order (smallest edge first) catches every possible box exactly once. Building the four boxes out of real cubes, or sketching them, makes the pattern obvious — the more the box looks like a cube, the tighter its skin. A second way of looking at it (changing focus from the outside faces to the hidden ones) explains why: the 12 cubes always contribute 72 faces in total, and every place where two cubes touch hides 2 of them, so the box with the most touching pairs has the least surface. That gives me an independent check on the list.
1STEP 1
Turn 'box made of 12 cubes' into 'three numbers that multiply to 12'
A box is three numbers multiplying to 12.
a × b × c = 12
Counting cubes in a box the way you count cubes in a crate — rows times columns times layers — is exactly the Grade 5 way of seeing volume, and it is what turns a stacking puzzle into an ordinary multiplication fact about 12.
5.MD.C.5Create A Physical RepresentationA box built from 12 cubes is exactly three whole numbers whose product is 12.
Why?
The cubes fill the box in rows, columns and layers, so the three edge counts multiply to the number of cubes.
🧱Every number has one prime factorizationKeep splitting a number into factors and you always land on the same bag of primes, no matter which split you tried first.
Why?
Each choice of one edge length hands over its partner factors, so working upward in order lists every triple with none missed.
🧱Divisors come in pairsEach divisor hands over its partner for free, so divisors pair off two by two unless one is its own partner.
2STEP 2
List every factor triple of 12, with no repeats and nothing missed
There are only four such triples.
12 = 1 × 1 × 12 = 1 × 2 × 6 = 1 × 3 × 4 = 2 × 2 × 3
Finding all the factor pairs of a number is a Grade 4 skill; doing it in a fixed order (smallest edge first, then second smallest) is what upgrades a pile of guesses into a proof that nothing was left out.
4.OA.B.4Make A Systematic List3STEP 3
Work out the surface area of each box on the list
Their surfaces are 50, 40, 38, 32.
S = 2(ab + bc + ac); 1×1×12: 2(1+12+12) = 50; 1×2×6: 2(2+12+6) = 40; 1×3×4: 2(3+12+4) = 38; 2×2×3: 2(4+6+6) = 32
The six faces of a box are just six rectangles, and a rectangle's area is length times width. Opposite faces are identical twins, so you only have to compute three rectangles and double the total.
6.G.A.4Draw A Diagram4STEP 4
Pick the smallest, and see why it is the boxiest box
The smallest is the boxiest one, 32.
50 ＞ 40 ＞ 38 ＞ 32
Once a list is known to be complete, choosing the smallest entry is just reading a short column of numbers — the thinking was all in making sure the column had every case in it.
7.G.B.6Make A Systematic List5STEP 5
Check it a different way: count hidden faces instead of outside faces
Counting joints also gives 32.
S = 6 × 12 - 2 × (touching pairs); 2×2×3: 72 - 2 × 20 = 32; 1×1×12: 72 - 2 × 11 = 50
Instead of counting what you can see, count what got hidden — that flips a hard shape question into an easy counting question, and it explains the pattern rather than just reporting it: pack the cubes as squarely as possible and you hide the most faces.
7.G.B.6Change Focus Count The ComplementAnswer
32 square inches
2 × 2 × 3 = 12
The units are right: an area answer in square inches, built from products of two edge lengths in inches. The magnitude is sensible too. The 12 loose cubes, if none of them touched, would show 12 times 6 = 72 square inches of face, so any stacked box must come in well under 72 — and 32 does. It must also be more than the surface of a solid cube of the same volume, and a 2 by 2 by 2 cube (only 8 cubes) already has surface 24, so 32 for a slightly bigger, slightly less cube-like solid is exactly the right neighbourhood. Finally, 32 is the smallest of the four values 50, 40, 38, 32, and the list of four is provably complete because it is the complete list of factor triples of 12.
Takeaway
Pack the cubes into the boxiest box you can — 2 by 2 by 3 — and the skin shrinks to 32 square inches, because squishing cubes together hides faces.
- Turn 'box made of 12 cubes' into 'three numbers that multiply to 12'
- List every factor triple of 12, with no repeats and nothing missed
- Work out the surface area of each box on the list
- Pick the smallest, and see why it is the boxiest box
- Check it a different way: count hidden faces instead of outside faces
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