187. Space objects evenly by adding “1”

[sossol]

There's a good reason algebra is the first advanced math subject taught after arithmetic. Algebra and the notation it uses is the language of all higher mathematics; if you don't understand it, you're not going to understand what follows. Get yourself an elementary algebra text and start reading it. Even if you don't finish the book, you'll have learned a bit about the notation and the rules that govern it.

Remember too that there are lots of people here, myself included, who can help you when you have questions or need clarification.

I appreciate the offer. I love math. I can read the formula well enough to recite it, but it turns to a jumbled mess in my brain when I try to solve. My attention deficit manifests in such a way that I can usually work out how a mechanism is made by watching it operate, but can't follow the math that it's based on. It makes car & appliance repair relatively easy for me, but diagnosis difficult.

Neil

[Make Things]

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.

[Make Things]

Another, related, example of Gauss' lateral thinking involves a formula for the sum of the numbers between n1 and n2 (where n1 > 1 and n2 > n1).

Now you can derive this formula using the above expression for the sum from 1 to n, although on your first try you'll probably make a very common mistake.

Gauss noted that the formula for a common average...

A = S / N

where:

A = average
S = sum of items
N = number of items

could be solved for the sum

S = A * N

and A and N are easily computed...

A = (n1 + n2) / 2
N = (n2 - n1 +1)

so:

S = (n1 + n2) * (n2 - n1 +1) / 2

which is identical to what you would get using the expression for the sum from 1 to n

It's funny, I can do physics like object moving in my head, as I stare into the corner of the room. I can know that something I build will work before I cut the first piece (I think they call it spatial mechanical thinking) but math is such a quagmire of numbers and letters that make absolutely no sense to me, no matter how hard I try to arrange things or how many times I'm told how it works. It's incredibly frustrating. I spent 40 years of my life thinking I was an idiot because I barely passed pre algebra, yet I can build things that will work 99% of the time.

[Make Things]

I appreciate the offer. I love math. I can read the formula well enough to recite it, but it turns to a jumbled mess in my brain when I try to solve. My attention deficit manifests in such a way that I can usually work out how a mechanism is made by watching it operate, but can't follow the math that it's based on. It makes car & appliance repair relatively easy for me, but diagnosis difficult.

Neil

Brother...maybe it's the water here in the Lafayette area. I share your pain!

EDIT: ADD my entire life. I'm awake at night because sounds and light are distracting to me. Oh, and now I'm battling chipmunks that are making home RIGHT under my office floor in my garage...so now my concentration level is even worse.

[mklotz]

Another instance where the " +1 " confusion arises is counting things that are spaced equally.

An example will help. How many drills are there in a standard American drill index - 1/16 to 1/2 by sixty-fourths ?

We can rephrase this problem by noting that 1/16 = 4/64 and 1/2 = 32/64. So, how many drills between 4 and 32 ? One is inclined to say, "Easy. There are 32 between 1 and 32 and four between 1 and 4 so 32 - 4 = 28 drills."

But that's wrong ! If you subtract 4, you're subtracting the 4/64 drill from the sum. You really need to subtract all the drills smaller than 4, which is 3 drills.

Then the correct answer is 32 - 3 = 29

When we generalize this (math alert here) we let 4 be n1 and 32 be n2 and the number of drills between them INCLUSIVE OF BOTH n1 AND n2 is given by:

n2 - (n1 - 1) = n2 - n1 + 1

and there's our " +1 ".

Having this formula in mind, we never need to think through the above again. We simply plug our numbers in and the correct answer falls out.

You'll note that I made use of that formula (without discussion) for "N" in post # 5 above.