Prior to BRILLIAthon's last problem, I had made a study about magic square connections.
Now this is for \(n \geq 2\) as \(n = 1\) yields no connections.
Imagine a magic square without numbers:
Now, in a magic square, you need to make sure all the rows, columns and (sometimes) diagonals add up to one number.
Most people would see connections in this picture. But I see differently. I see connections:
Why ? Why not see as everybody else?
Well, from what normal people see, the amount of connections in a magic square forms this arithmetic sequence:
There's no th term.
And there is additional connections.
But, from my perspective:
There's a th term for this sequence: .
And there is no additional connections.
Also, when , it gets harder to draw a magic square with all the connections, so we need an algebraic approach.
What people normally see is (excluding the ).
However, the ratio shows, the amount of magic square connections missing gets larger with each increase of to .
Also, the amount of triangles increases at a geometric rate:
In my square above, you should see (forgive me for the messy lines - it was done on PowerPoint) triangles. (If you can't see triangles, don't worry - I first did this study on paper, so I know exactly how many are there.)
Continuing the sequence in both directions, we obtain:
The th term is: (oddly, or don't match the exact sequence.) .
Note that this th term is from my perspective.
Now, imagine functions and (where denotes the magic square, denotes area and denotes perimeter):
The ratios are:
Based on these facts:
Now, if we divide the amount of triangles by the amount of connections ( for short), we get:
Inverting it gives us the amount of connections divided by the amount of triangles:
Now, showing a square and increasing by gives us this:
The th term for this sequence is - this th term denotes the function (where is the amount of dots.)
Also, since a square's rotational symmetry is , a magic square with all the connections will also have a rotational symmetry of .
Now all of this information is useful for solving questions like this:
You have a square.
You have one clue:
Find the true amount of connections, the amount of dots, the amount of triangles and add them together.
Now, to the last BRILLIAthon problem:
How many connections are there in a square:
I hope you enjoyed this study and learnt something from this!