  Calculation Challenge Game: Largest and Smallest Number of Valid Solutions
 With the Goal integer = 0 and factors 0, 0, 0, and 0, there are 3240 valid solutions.  There are no valid equations that don’t equal 0; the remainder of the 7680 possible equations are all divide-by-zero invalid equations. That’s the largest number of valid solutions for any Goal integer and set of factors that I’ve come up with so far.  Can you think of a combination of a Goal integer and factors that has more valid solutions? As an aside, if the Goal integer = 24 and the factor range is 1 to 13, the largest number of valid solutions is 744 and occurs for only 1 factor set:  12, 12, 12, 12.
 Largest number of valid solutions    SaneSchool
 Excellent work done quickly.    With the Goal integer = 24 and factors 1, 4, 5, and 6, there is only 1 valid solution: 6 / ((5 / 4) — 1) Wow, that one’s pretty difficult!  In fact, given the Goal integer = 24 and a factor range of 1 to 13, there are only 2 other sets of factors that have only one valid solution: 1. Factors = 1, 3, 4, 6 2. Factors = 2, 3, 5, 12 Can you come up with the one valid solution for each of those cases? One is the smallest non-zero number of valid solutions.  Of course, there are many combinations of a Goal integer and factors that result in zero valid solutions (for example, Goal integer = 24 and factors of 8, 11, 11, and 13), but we only want the smallest non-zero number of valid solutions here.
 Smallest (non-zero) number of valid solutions 