Quote Originally Posted by mklotz View Post
The " + 1 " appears automatically if you think about how many spaces you need to create.

If you arrange N screws in a row you automatically create N - 1 spaces. Two screws create one space between them, three screws create two spaces, four create three, and so on. However you need to add the two spaces at the ends, one between the leftmost screw and the leftmost edge of the board, one between the rightmost screw and the rightmost edge of the board. Thus, you need

N - 1 + 2 = N + 1

spaces and voilà, there's your " + 1 ", and dividing N + 1 into the length of the board will give you the size of the required space(s).

Note that this formulation works for the case of only one screw, yielding a space equal to half the width of the board, i.e. put the screw in the middle of the board.

It also works for zero screws, yielding a spacing equal to the board width.

While the latter two cases aren't of practical interest, they're mathematically satisfying showing the formulation would work for all possible values of N.
I knew there was a good formula to explain it...heck I searched for a method for a long time (many, many years ago). I stumbled across the idea by accident once and was so happy it worked I didn't think much of it but didn't want to jinx myself by thinking I'd finally solved the problem...kind of like, if I look into this too deep, I'll find that my return wasn't as perfect as I thought it would be.

I'm not even kidding...most of these little tips are from trial and error, and if I'm being honest, just dumb, stupid luck. Tips become fun when there's a problem that you find a solution to, and this one, literally took me 10 years to really peg down. I am happy to see there's an even better way to explain this than me just getting lucky with the solution.