Since uncountably many problems have unique right answers, I'd guess the cardinality of the set of problems with unique right answers is the same as the cardinality of the set of problems without. :)
There are infinitely many addition problems involving two numbers, and that's a subset of the set of problems that the article claims to be smaller.
> a question that a mathematician might ask
I agree that this is not a very useful definition, by virtue of its extreme and probably excessive inclusiveness, but I think that it's hard to do any better without using words like 'interesting' that themselves need careful definition. (It's fair to argue that my definition in turn requires clarification of the term 'mathematician', but I can weasel my way around that by replacing it with 'person', or else just declaring that anyone interested in trying to ascertain the normality of a number is mathematically minded enough to be called a mathematician.)
By this definition, I think that it is hard to argue with my claim to have produced an uncountable family of problems—simply because, at least classically, to do so you'd have to produce a specific number x about whose normality no mathematician could ever ask. I could then demolish that counterexample by asking you if that particular number x was normal. :-)
For example, why is the set of all questions utterable by humans not a countable set? By that reasoning there are more numbers than questions!