MATH or MAGIC!

Extrapolating on the principles used for squaring, to get the cube of a number (of two or more digits), simplify by splitting the number into two parts, a and b.

Thus (a + b)³ = a³ + 3a²b + 3ab² + b³

The solution comprises four parts, neatly fitting the four boxes shown above. Just adjust for excess carry over.

1. a³
2. a² x b x 3
3. b² x a x 3
4. b³

A simple example to illustrate this technique.


23³ = 12167

The steps are:

1. 3³ = 27, put down the 7 in the rightmost box and carry over the 2
2. 3² x 2 x 3 = 54, plus the 2 carried over is 56, put down the 6 and carry over the 5
3. 2² x 3 x 3 = 36, plus the 5 carried over is 41, put down the 1 and carry over the 4
4. 2³ = 8, plus the 4 carried over, is 12 in the leftmost box

Another example.


108³ = 1259712

The steps are:

1. 8³ = 512, put down the 2 in the rightmost box and carry over the 51
2. 8² x 10 x 3 = 1920, plus the 51 carried over is 1971, put down the 1 and carry over the 197
3. 10² x 8 x 3 = 2400, plus the 197 carried over is 2597, put down the 7 and carry over the 259
4. 10³ = 1000, plus the 259 carried over, is 1259 in the leftmost box

That's a million-dollar figure worked out manually!