Hello rainer

I'm not sure that I agree with your working so far - that is, if I'm understanding your requirements.

Take a look at the diagram I've attached. Originally Posted by

**rainer** Every corner of the cube must have a line connecting it to every other corner. I am interested in the sum of the length of these lines.

In one cube the sum, if I'm not mistaken, is:

I agree with this. This sum can be thought of as:(1) The sum of the lengths of the lines lying in the plane ; PLUS

The additional lengths created when the plane is added; which in turn is comprised of: (2) The sum of the lengths of the lines in the plane (the same as (1)); PLUS

(3) The sum of the distances of each of the points in plane #1 from each of the points in plane #2.

This last sum, (3), can be found by taking the sum of the distances of one of the points in plane #1 - say, - from each of the points in plane #2, and multiplying by 4 (since there are four points in plane #1, each indistinguishable from each other). This sum (3), then, is .

Thus, when we add the totals (1), (2) and (3) together, we get:
Now, when I place another cube right smack alongside the first cube, there are now 12 planets connected by lines of force.

The sum of these lines, if I'm not mistaken, is:

If I understand you correctly, I think you are mistaken here.

When we add four more 'planets' in plane #3, we get, in addition to the distances above:(4) The sum of the lengths of the lines in plane #4; this is, of course, the same as (1) and (2); namely . PLUS

The sum of the all the distances of from each of the points ; which comprises:(5) the distances of the points in plane #2, which is the same as (3) above; namely ; and

(6) the distances of the points in plane #1, which is (taking the distances from and multiplying by 4) .

So I reckon that the total for the 12 planets is:To see the pattern being created here, break the expression above into three components:(1) for plane #1

(2) + (3) for plane #2

(4) + (5) + (6) for plane #3

and notice that (1), (2) and (4) are the same; (3) and (5) are the same; (6) is the new expression formed from the distances of the points in the first plane from the last. So we can re-arrange these as:i.e.So when the next new plane is added comprising points , there will be additional totals equal to:(4) + (5) + (6) for the points in planes #2, #3 and #4; PLUS

(7) the distances from plane #1 to plane #4; namely .

So this, when re-arranged, is equivalent to:... and so on. I hope you can see the pattern emerging. I realise that I'm still quite a way short of developing a formula for the total with planes, but at least you should be able to see how to continue - if you still want to, that is!

Lastly:
AND: In this scheme of things, the cubes line up in a straight line. Would the sum of lengths be the same if the cubes were arranged differently (but always so that the xth cube is added in such a way that four of its corners line up exactly with the four corners of one of the other cubes)?

Thanks

No, I'm sure that the sum *won't* be the same if the cubes are not arranged in a straight line.

Grandad