# Maths Puzzle

• January 1st, 2010, 05:40 PM
Maths Puzzle
Hi all,

Its just a little problem that I thought up by myself that I couldnt really develop it beyond a few simple steps.

Basically I wanted to know if a function existed or could be made that would help to handle this question.

I wanted to know what structure (in a 3 dimensional field), could be build to get the most lines per dot as part of the structure but you can only include the lines on the outside of the shape.

I got this:

0 dots = 0 lines
1 dot = 0 lines
2 dots = 1 line
3 dots = 3 lines (triangle)
4 dots = 6 lines (as far as I can tell a pyramid is the most efficient for 4 dots)
5 dots = 9 lines (2 pyramids on top of each other, sharing a base)

Its just guess work no paper used.

But I am curious about the possibilities and if a formula exists that would make this simpler?

Thanks for the responses in advance.
• January 2nd, 2010, 12:14 AM
smokey
I would think 5 dots = 10 lines, you have a new line to the 4 dots from the lat 'shape'
plus the 6 lines already there.
So it goes 1 3 6 10
I would suggest the next number is 15, one extra line from the new dot to the other 5 dots. Hence 21 for 6 28 for 7?
1 3 6 10 15 21 28 36 45 55 67...?
All you can do is add a line form each existing dot to the new dot, all the other dots
are joined.
• January 2nd, 2010, 04:45 AM
Quote:

Originally Posted by smokey
I would think 5 dots = 10 lines, you have a new line to the 4 dots from the lat 'shape'
plus the 6 lines already there.
So it goes 1 3 6 10
I would suggest the next number is 15, one extra line from the new dot to the other 5 dots. Hence 21 for 6 28 for 7?
1 3 6 10 15 21 28 36 45 55 67...?
All you can do is add a line form each existing dot to the new dot, all the other dots
are joined.

Im not sure this is correct, I was only counting lines on the outside of the figure, U cant include lines on the inside.

Basically what shape imagine if its filled in once its complete has the least number of dots added and the most number of sides, the lines between dots. (In other words the edges, I dont see how 10 lines is possible with only 5 dots)

Maybe I havent explained it properly, Its really early in the morning and I just woke up so can someone help explain it further if this above is correct
• January 2nd, 2010, 06:29 AM
wallaby
I think you explained the problem just fine. As far as a relationship between the dots and the lines goes i got,

#lines = 3* (#dots) - 6

However this is only for more than 3 dots, which to me makes sense. Since the 3d shapes we are constucting are a collection of intersecting planes and since we need at least 3 points to construct a plane in 3 dimensional space we should therefore start with 3 dots.
• January 2nd, 2010, 10:08 AM
Guitarist
Always: If I read you correctly, you have re-invented one of the most famous math puzzles of all time: it's called something like "the Konigsberg Bridge" problem. Look it up, if you want.

Anyway, assuming, as you seem to be suggesting, that no lines may intersect (cross each other), then the solution had to wait until the great Leonard Euler (say "oiler") turned his mind to it.

Supposing that "D" is the number of dots, "L" is the number of lines, and that "R" is the number of enclosed regions (i.e.those that are bounded by non-intersecting lines) he gave us the formula

D + R - L =1. For small D this is easy enough to see, for larger D it may not be

I have never seen a proof of this, but I have no doubt it would be inductive.

Nice work
• January 2nd, 2010, 10:59 AM
Quote:

Originally Posted by Guitarist
Always: If I read you correctly, you have re-invented one of the most famous math puzzles of all time: it's called something like "the Konigsberg Bridge" problem. Look it up, if you want.

Anyway, assuming, as you seem to be suggesting, that no lines may intersect (cross each other), then the solution had to wait until the great Leonard Euler (say "oiler") turned his mind to it.

Supposing that "D" is the number of dots, "L" is the number of lines, and that "R" is the number of enclosed regions (i.e.those that are bounded by non-intersecting lines) he gave us the formula

D + R - L =1. For small D this is easy enough to see, for larger D it may not be

I have never seen a proof of this, but I have no doubt it would be inductive.

Nice work

Yeah my idea was that lines couldnt intersect as that would be another dot at the place of intersection.

My original thoughts were very different to a problem related to bridges

My original thoughts were something along the lines of my believe that Nature is really efficient in its mathematical structures that it tends to build and I wanted to know what would the outside look like if u had some structure that was continuously having mass (my dots) added to it a little at a time.

I know it seems unrelated but I guess one thought leads to another, and u end up with odd questions like this :)

So I needed to try find out what is the most efficient shape that u can attain from the outside perspective

I still dont know what sort of shape this formula builts or if its always changing

One of my friends said it would be a star like shape with points sticking out everywhere, but I am not so sure

The other thing we thought of was was a curve can be described as a series of infinite straight lines which I thought would also have infinite points, but that was just a thought and I didnt think that was the most efficient

anyway thanks for the answer, If u know what sort of structure it would build, could u elaborate as I am curious...or does the shape not matter that much for this question.

EDIT: my physics knowledge is really only basic and I was just thinking about the relationship between mass and energy and if mass can be converted to energy, I thought it might be possible for energy (or dark energy) to convert itself slowly into mass, (this added mass over time would change the actual mass and would create a discrepency between the actual mass of a body e.g. galaxy and the mass that we see or the visible part of it)where ever dark matter is I then thought what kind of effects this would have on the structure of the universe and it might just be enough to change the way the universe looked and behaved making some of the current theories a bit more interesting.