How many different problems are possible in each generator?

The number of different problems for most generators is well in the thousands, and some are even in the tens or hundreds of thousands. However, these large numbers primarily result from the fact that each coefficient is a randomly selected number within a given range. So in linear equations, for instance, even though there are only 42 base forms, the possibilities for each of the coefficients in these equations multiply to produce large numbers like these. Additionally, the number of possible problems depends heavily on the settings. Certain settings configurations can result in only a few hundred possible problems, while others (although not necessarily practical for teaching purposes) can result in many more.

How are the answers to the problems verified?

For many of the generators, the answer to the question is actually determined first and is used to build the rest of the problem. These all follow a process of finding the generalized solution to the problem, then using that solution with specific constraints. So if we find the general solution to a particular linear equation (for example), we can then put constraints on that solution (maybe forcing it to be an integer), and use that to figure out what the coefficients must be. However, for generators that don’t follow this pattern, the answer is calculated dynamically as the problem is generated. Also, all of the problem types have been verified extensively with a computer algebra system (CAS) by matching the answer supplied by the generator to the one calculated by the CAS. Each generator has been tested over 150,000 times.

How do the settings work?

The settings mainly work by limiting particular random aspects of problem generation. To give an example, before settings, all of the numbers in addition and subtraction were integers between -10 and 10. However, with settings, one can now create their own range with integers anywhere from -999 to 999 (for example, [-999, 999], [0, 5], [10, 20], and even [1, 1] are all valid ranges), and the coefficients will be selected from that range. Similarly, you can choose a specific form (or multiple) in factoring and quadratics, and the generator will randomly create a problem within that form. As a whole, the purpose of the settings is to enable users to generate problems with specific mathematical characteristics while maintaining as much randomness as possible.