Start out simple. Write a function that iterates through numbers between 0 and 500, and does the sum the cubes test. Usually we just call this type of algorithm a search over parameter space. For all sorts of problems for which we have no answers, we simply search for it.

for (hundreds_digit from 0 to 5)
for (tens_digit from 0 to 9)
for (ones digit from 0 to 9)
[B]if (sum the digits^3) == number[/B]
print number

- or - more generally for an arbitrary base:

You could do this on Decimal (base 10) or even Hex (base 16)

for (hundreds_digit from 0 to 5)
for (tens_digit from 0 to Base-1)
for (ones digit from 0 to Base-1)
// be careful about the 'sum the digits^3'
// make sure you know what base you are in.
// you may want to convert the number to decimal first
[B]if (sum the digits^3) == number[/B]
print number

Armstrong numbers aren't a solved problem, so you aren't going to find a closed-form formula to generate the next number. You can probably find a more clever way to perform the search over parameter space of numbers. I highly encourage you to try to make a more efficient program; it'll train your critcal thinking skills.

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