Writing a C++ version of the algebra game 24

Question

I am trying to write a C++ program that works like the game 24. For those who don\'t know how it is played, basically you try to find any way that 4 numbers can total 24 th

Answer 1

You can just use Reverse Polish Notation to generate the possible expressions, which should remove the need for parantheses.

An absolutely naive way to do this would be to generate all possible strings of 4 digits and 3 operators (paying no heed to validity as an RPN), assume it is in RPN and try to evaluate it. You will hit some error cases (as in invalid RPN strings). The total number of possibilities (if I calculated correctly) is ~50,000.

A more clever way should get it down to ~7500 I believe (64*24*5 to be exact): Generate a permutation of the digits (24 ways), generate a triplet of 3 operators (4^3 = 64 ways) and now place the operators among the digits to make it valid RPN(there are 5 ways, see Omnifarious' answer).

You should be able to find permutation generators and RPN calculators easily on the web.

Hope that helps!

PS: Just FYI: RPN is nothing but the postorder traversal of the corresponding expression tree, and for d digits, the number is d! * 4^(d-1) * Choose(2(d-1), (d-1))/d. (The last term is a catalan number).