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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 units³.

How many cuboids exist for which the volume is less than 100 units³ and the integer side lengths are in an arithmetic sequence?

Solution

Although 333 = 27 is a cuboid with a volume less than 100 units³, 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 units³?
Can you generalise for a volume less than V units³?