## Arithmetic Volume

#### Problem

A sequence is arithmetic if the numbers increases by a fixed amount. For example, 2, 5, 8, are in an arithmetic sequence with a common difference of 3, and if these numbers represented the side lengths of a cuboid, the volume, V = 258 = 80 units^{3}.

How many cuboids exist for which the volume is less than 100 units^{3} and the integer side lengths are in an arithmetic sequence?

#### Solution

Although 333 = 27 is a cuboid with a volume less than 100 units^{3}, we shall discount cubes on the grounds that 3, 3, 3, is a trivial example of an arithmetic sequence.

By considering the first term, `a`, and the common difference, `d`, we have a method of systematically listing all of the solutions:

`a`=1, `d`=1, 123 = 6`a`=1, `d`=2: 135 = 15`a`=1, `d`=3: 147 = 28`a`=1, `d`=4: 159 = 45`a`=1, `d`=5: 1611 = 66`a`=1, `d`=6: 1713 = 91`a`=2, `d`=1: 234 = 24`a`=2, `d`=2: 246 = 48`a`=2, `d`=3: 258 = 80`a`=3, `d`=1: 345 = 60

Hence there are exactly ten cuboids.

What is the maximum volume obtainable for a cuboid with side lengths in an arithmetic sequence and having a volume less than 1000 units^{3}?

Can you generalise for a volume less than V units^{3}?