Number pyramid

We get $(a+b+c+d)+(b+e+f+g)+(d+g+h+i)=3N=0+1+2+3+4+5+6+7+8+9+b+d+g=45+b+d+g$ .

So $b+d+g=3(N-15)$ .

Since $b, d, g$ are different digits, $6\leq b+d+g\leq 24$ , $2\leq N-15\leq 8$ , $17\leq N\leq 23$ .

So the minimum of $N$ is $17$ , and here is a possible numbering:

So the maximum of $N$ is $23$ , and here is a possible numbering: We get from these that $y-x=23-17=\boxed{6}$ .

For the sum to be maxed the overlapping triangles should contain 7,8 and 9. So the sums of the 3 triangles is now 15,16 and 17. To make the sums equal the only pairs of the remaining 6 numbers that work are: 5,3; 6,1 and 4,2. This leads to 23. In the same way you will find a minimum sums of 17 when starting with 1,2 and 3 in the overlapping triangles.