A Square and a Cube: Thinking Puzzles

Thinking puzzles on squares and cubes: prime factors decide them, number patterns predict them, and painted cubes are counted part by part.

Key ideas

A number is a perfect square when every prime in its factorisation has an even exponent, and a perfect cube when every exponent is a multiple of 3.
To make n × k a perfect cube, multiply in just enough of each prime to raise its exponent to a multiple of 3.
To make n ÷ k a perfect square, divide out one of each prime that has an odd exponent.
The cube root of a perfect cube can be read off: its units digit from the cube’s units digit, its tens digit from the thousands.
Number patterns give shortcuts: odd numbers add up to squares, and cubes add up to squares of triangular numbers.
A number machine (square, add, multiply) is an algorithm: do its steps in order.

Remember

Smallest k for a cube	72 = 2³ × 3², so k = 3: 216 = 6³
Smallest k to divide for a square	72 ÷ 2 = 36 = 6²
Odd numbers	1 + 3 + 5 + … + (2n − 1) = n²
Cubes	1³ + 2³ + … + n³ = (1 + 2 + … + n)²
Whole numbers strictly between n² and (n + 1)²	2n
Both square and cube	sixth powers: 1, 64, 729, …
Cube root units digit	2 ↔ 8 and 3 ↔ 7 swap; 0, 1, 4, 5, 6, 9 stay

Watch out

Multiplying by the whole leftover prime power instead of only what is missing.
Saying 8 cubed ends in 8: 8³ = 512 ends in 2.
Counting n² and (n + 1)² themselves as “between” them.

Worked example

A cube of side 10 cm is built from 1,000 unit cubes, and its outside is painted. How many unit cubes have (a) three painted faces (b) exactly two (c) exactly one (d) none?

Working: corners 8; edges 12 × 8; faces 6 × 8²; inside 8³; check: 1000 = 10³

Answer: (a) 8; (b) 96; (c) 384; (d) 512