Determine the number of integer solutions to:
|x|+ |y| + |z| = 15
Note:
The absolute value function F(x) = |x| is defined as:
x if x ≥ 0
F(x) =
-x if x < 0
Let's solve the general case.
|x| + |y| = n geometrically is the number of integral points on the edge of a diamond whose 4 vertices are (0,+/- n) and (+/- n, 0).
This equals 4n, unless n = 0, in which case it equals 1.
|x| + |y| + |z| = n geometrically is the number of integral points on the surface of an octahedron whose 6 vertices are (0,0,+/- n) and (0, +/- n, 0) and (0,0, +/- n).
Let's count these two different ways.
Method 1: let's slice the octahedron parallel to the xy plane into one slice of for each possible z value (from -n to n) and sum them up.
Total points, if n > 0 = 1 + 4*1 + 4*2 + ... + 4(n-1) + 4(n) + 4(n-1) + ...+ 4*2 + 4*1 + 1.
this equals 2 + 8*(1 + 2 + ... + (n-1)) + 4n =
2 + 8*(n-1)*n/2 + 4n = 4n^2 + 2. This is a very cool result.
When n = 15, this equals 902, which is Charlie's number, so I assume that we were both right.
Method 2: Count the points on the octahedron directly.
First, count points on the 8 triangular faces, not counting the edges.
These equal 8*(1 + 2 +...+ (n-2)) = 4n^2 - 12n + 8
Next, count points on the 12 edges, not counting the vertices.
this is 12*(n-1) = 12n - 12
Next count the 6 vertices.
Total = (4n^2 - 12n + 8) + (12n - 12) + 6 = 4n^2 + 2.
Of course, this is only valid if n > 0
Full general case:
n = 0 Solutions = 1
n > 0 Solutions = 4n^2 + 2
General Solution -- Two geometric methods
