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A good consultant doesn't
No. In most cases consultants aren’t creating problems, they’re looking for them in the same way a mathematician would so they can create a solution. Hell in a lot of cases, clients reach out to us with their problems.
Nay
...
Picture is more relevant than the point OP was trying to make
Engineer, not mathematician. Tho an engineer is likely both?
An engineer will know Math related to his or her field that enables them to to apply it to a real world problem that they're trying to solve for. I'm talking of a true engineer here such as an aerospace engineer etc rather than an EE grad that is doing consulting for a living. A mathematician may not be as interested in a real life problem as being able to prove some theory and appreciate the beauty of the proof whether it be applicable in the real world or no. A mathematician will also likely have a more broader knowledge of the world of Math than an engineer who's restricted to the Math required for his or her field (an EE grad will know a lot more about complex numbers and calculus , for example, than will a Mech engineer who will know a lot more about Arithmetic and classical Math, for example)
Just my opinion
Solve the immediate problem but uncover many others that can be solved
A great consultant starts with a solution, creates a problem, and wins new engagement by providing a solution for said problem 💰