if factorials can be used.
Any time I see a question like this: http://www.mathhelpforum.com/math-he...make-24-a.html I wonder:
Of the possibilities of digits, how many can be solved by the four simple operations, how many can be solved if we allow other symbols, and how many are unsolvable?
I can't come up with any way to google an answer, though I'm sure someone must have done this before. Any interest?
I'm confident that (1, 1, 1, 1) cannot be made into 24, regardless of operations.
Anyone want to join in to help with the other 6560 possibilities?
I've done this, limiting the operations to the four standard +, -, *, /. I do not have the results where I can get to them easily, and I'm not even sure I saved the files.
However, I would like to point out another use for this type of analysis. By counting the number of solutions for each set of 4 numbers you can rank the problems by difficulty. A problem with many solutions is easier than one with few.
There are also a couple of fine points to consider: Do you allow intermediate steps which result in negative numbers? And how about fractions-- must each step yield an integer result? My understanding, based on heresay, is that these problems ("X24") are used to drill arithmetic in elementary schools, and depending on how how much math the kids know, problems involving fractions or negative numbers might be considered to be too hard.