Dose the word "least" have the meaning of "rigorous"

Having lived in the central Australian desert I can tell you there are a few problems.

But to get to your question. The sentence is clumsy, from a native English speaker's point of view. What the sentence is trying to convey is that obtaining water is not the most difficult aspect of living in the desert. But it doesn't say it clearly or plainly.

"Rigorous" has a meaning related to difficulty or struggle. If I undertake rigorous exercise it means that it was fairly demanding. If I say "there was a rigorous inspection of my office" I mean that it was very detailed and painstaking. Rigorous can also mean "tough".

The word "least" is usually used when making comparisons. If I say that "finding a house is the least of my worries" it means that finding a house is not really much of a problem for me and that I have other things to worry about that are more difficult.

I hope that helps. Others may have better examples.

The sentence would be more easily understood if "of them" were added at the end.

No, the sentence means that obtaining water is not the most MINOR problem of living in the Australian desert.

The main parts of the sentence could be rephrased: "Not the least of the problems... is obtaining water." Does that help to clarify that obtaining water is an important problem? "Not the least" is idiomatic, and generally means "fairly important."

So your teacher has it backwards -- obtaining water IS a "rigorous" or "difficult" part of living in the desert (although, as goodfielder says, that's not a common way to use the word rigorous).

Wy explained this correctly, but in case the explanation is confusing, here is another way to look at the sentence. If finding water was the least of the problems, then it would be the smallest (or easiest) problem, but since it is not the least of the problems, it would not be a small problem. By not simply saying that it is a big problem, the sentence is an example of what is known as an understatement.