N - M - x -1 and N -x - 1 -M
These are the same. And if I substitute for N
M + x + 3 - x - 1 - M = 2 for all N, M and x.
In the same way you can prove that the right hand cross differences are equal to 1 and the bottom ones are also 1.
So 211 is the result for every set of N, M and x.
Bob
]]>With that equation there are quite a few solutions. So it's not so remarkable to find one with other interesting properties. When I get a moment I'll see if I can add a second constraint based on what I'll call the cross-differences. I've just got to find a pencil, some paper and some time.
Bob
]]>Welcome to the forum.
Great diagrams. But what are we supposed to be noticing? Are you able to post some more examples?
LATER EDIT: As my morning has gone on I've given this more thought. Can I make my own example?
Just choosing numbers at random, the answer was no. So then I thought I'd try some algebra.
In place of the top numbers, 9 and 2 I put N ad M. In place of the bottom addition number (4) I put x. Then I developed the triangle according to what I think are the rules.
So bottom left I calculated N - x - 1 and M + x + 1.
Bottom right I had N - x - 2 and M = x + 2
Across the bottom I formed equations by requiring that x is the add number.
N - x - 1 + x = M + x + 2
and
N - x - 2 + x = M + x + 1
Fortunately these both simplify to the same equation:
N = M + x + 3
So now I'm in a position to construct my own example. I want N, M and x all to be whole numbers under 10 so I chose M = 1 and x = 3.
That requires N = 1 + 3 + 3 = 7 and the numbers at the vertices are 71, 35 and 62. And that's as far as I've progressed.
Bob
]]>Question:
- What do you call these number connections?
- Has someone ever found this triangle connection before?
Thanks in advance.
Ferry