A Square and A Cube

1. The story: Queen Ratnamanjuri's 100 lockers

Queen Ratnamanjuri left her son Khoisnam and $99$ relatives a puzzle. In a secret room are 100 lockers, all closed. Each person ($1$ to $100$) takes a turn:

• Person 1 opens every locker.
• Person 2 toggles (open↔close) every 2nd locker — $2,4,6,\dots$
• Person 3 toggles every 3rd locker — $3,6,9,\dots$, and so on up to Person 100.

A locker ends open only if it is toggled an odd number of times. Locker $k$ is touched once by each person whose number is a factor of $k$. So the question becomes: which numbers have an odd number of factors?

2. Why squares have an odd number of factors

Factors come in partner pairs whose product is the number. For $6$: $1\times6$ and $2\times3$ — factors $1,2,3,6$, an even count ($4$). For $9$: $1\times9$ and $3\times3$ — here the pair $3\times3$ is a repeat, so the factors $1,3,9$ are odd in count ($3$).

A perfect square always has one "middle" factor that pairs with itself (the square root), leaving the total count odd. Every other number pairs all its factors off neatly, giving an even count.

The passcode clue: "the first five lockers touched exactly twice". Touched twice = exactly two factors = a prime ($1$ and itself). The first five primes give the code 2-3-5-7-11.

3. Square numbers (Section 1.1)

The area of a square of side $n$ units is $n\times n$ unit squares — that is why $1,4,9,16,\dots$ are called squares. We write $n\times n=n^{2}$, read "$n$ squared".

Squares work for fractions and decimals too: $\left(\tfrac35\right)^{2}=\tfrac{9}{25}$ and $(2.5)^{2}=6.25$. Squares of natural numbers are called perfect squares ($1,4,9,16,25,\dots$).

4. Patterns and properties of perfect squares

(a) Units digit. Every perfect square ends in $0,1,4,5,6$ or $9$. None end in $2,3,7$ or $8$.

A number ending in $1$ or $9$ has a square ending in $1$ (e.g. $11^{2}=121,\,19^{2}=361$). A number ending in $4$ or $6$ has a square ending in $6$ (e.g. $4^{2}=16,\,6^{2}=36,\,14^{2}=196$).

(b) Trailing zeros. The number of zeros at the end of a square is always even (double the zeros of the base): $10^{2}=100$ (2 zeros), $100^{2}=10000$ (4 zeros). A number with $3$ zeros has a square with $6$ zeros.

(c) Parity. Square of an even number is even; square of an odd number is odd.

(d) Sum of odd numbers. Adding consecutive odd numbers starting from $1$ builds the squares:

Test by repeated subtraction: keep subtracting $1,3,5,7,\dots$ from a number. If you reach exactly $0$, it is a perfect square; if you cross below $0$, it is not.

(e) Triangular numbers. Two consecutive triangular numbers add to a square: $1+3=4=2^{2}$, $3+6=9=3^{2}$, $6+10=16=4^{2}$.

5. Square roots

If the area of a square is $49$, its side is $7$ because $7^{2}=49$ — we call $7$ the square root of $49$. In general, if $y=x^{2}$ then $x=\sqrt y$.

Every perfect square has two integer square roots, one positive and one negative, since $8^{2}=64$ and $(-8)^{2}=64$.

Finding a square root by prime factorisation. A number is a perfect square if its prime factors can be split into two equal groups (i.e. every prime occurs an even number of times / can be paired). The product of one group is the square root.

6. Estimating square roots

For a number that is not a perfect square, trap it between two known squares.

For a perfect square like $1936$: it lies between $40^{2}=1600$ and $50^{2}=2500$, and ends in $6$, so the root ends in $4$ or $6$ — it is $44$ or $46$. Since $45^{2}=2025>1936$, the root is below $45$, so $\sqrt{1936}=44$.

7. Cubic numbers (Section 1.2)

A cube of side $n$ is filled by $n\times n\times n=n^{3}$ unit cubes — a $2$-cm cube needs $8$ unit cubes, a $3$-cm cube needs $27$, a $4$-cm cube needs $64$. So $1,8,27,64,125,\dots$ are perfect cubes.

Cubes work for fractions/decimals/negatives: $\left(\tfrac46\right)^{3}=\tfrac{64}{216}$, $(13.08)^{3}=2237.810112$, $(-6)^{3}=-216$. The cube of an odd number is odd; the cube of an even number is even.

Consecutive odd numbers build cubes too: $1=1^{3}$, $3+5=8=2^{3}$, $7+9+11=27=3^{3}$, $13+15+17+19=64=4^{3}$, $21+23+25+27+29=125=5^{3}$.

Taxicab / Hardy–Ramanujan number: $1729$ is the smallest number expressible as a sum of two cubes in two ways: $1729=1^{3}+12^{3}=9^{3}+10^{3}$.

8. Cube roots

Since $8=2^{3}$, we say $2$ is the cube root of $8$, written $\sqrt[3]{8}=2$. In general, if $y=x^{3}$ then $x=\sqrt[3]{y}$, and $\sqrt[3]{n^{3}}=n$.

Key fact: each prime factor of a number appears three times in its cube. So a number is a perfect cube if its prime factors split into three identical groups (every prime occurs a multiple of $3$ times).

If the primes cannot be grouped into triplets, the number is not a cube: $500=2\times2\times5\times5\times5$ — the two $2$s are left over, so $500$ is not a perfect cube.

9. Successive differences & a pinch of history

Squares: first differences are the odd numbers $3,5,7,9,\dots$; the second differences are all equal to $2$.

Cubes: taking successive differences of $1,8,27,64,125,216$ becomes constant only at the third level (all $6$). The first known lists of squares and cubes were made by the Babylonians around $1700$ BCE. In Sanskrit, varga meant square and ghana meant cube; mula (root of a plant) gave us "root".