TITLE: Can you generate math problems that are solveable?
QUESTION [1 upvotes]: If you take Linear Programming, it problems are formulated like this:

You know that Cabinet X costs 10 cents per unit, requires 6 square feet
of floor space, and holds 8 cubic feet of files.
Cabinet Y costs 20 dollars per unit, requires 8 square feet of floor
space, and holds 12 cubic feet of files.
You have been given 140 dollar for this purchase, though you don't have
to spend that much. The office has room for no more than 72 square feet
of cabinets.
How many of which model should you buy, in order to maximize storage volume?

But you notate it as a math formula, and not in words. Now suppose I replace all the values in this formula(10, 6, 8, 20, 12, 140, 72) with random other numbers, so I create a completely new problem.

Do they take longer to solve, than to verify the solution?
Can I check that they are solveable without solving them?
Can I check how difficult they will be to solve, without solving them?

REPLY [1 votes]: All problems of this structure are solvable.  You can find the maximum number of X cabinets by checking each constraint.  In your example, there is room for $12$ of them, so that is the maximum.  Similarly, there is a maximum of $9$ Y cabinets.  Now you can just loop over $x$ from $0$ to $12$ and $y$ from $0$ to $9$,check whether you can fit $x$ type X cabinets and $y$ type Y cabinets, if so compute the volume, and report the optimum result.  
In your example it is clear that Y cabinets are more efficient than X cabinets and we are able to use all the floor space with Y cabinets.  Also the cost is so low that it is not a constraint.  This makes the problem easy.  You want conflicting desires to make it harder.  If you raised the prices dramatically, so you couldn't afford all the Y cabinets you want, it would be harder.
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