smc

We declare a point $(x,y)$ to make up for the extra steps that the bug has to move. If the point $(x,y)$ satisfies the property that $|x-3|+|y+2|+|x+3|+|y-2|\le 20$, then it is in the desirable range because $|x-3|+|y+2|$ is the length of the shortest path from $(x,y)$ to $(3,-2)$ and $|x+3|+|y-2|$ is the length of the shortest path from $(x,y)$ to $(-3,2)$. If $-3\le x\le 3$, then $-7\le y\le 7$ satisfy the property. there are $15\times 7=105$ lattice points here. else let $3<x\le 8$ (and for $-8\le x<-3$ because it is symmetrical) We set 8 as the upper bound for x because the shortest distance from $(-3,2)$ to $(x,y)$ added to the shortest distance from $(x,y)$ to $(3,-2)$ is $|x-3|+|y+2|+|x+3|+|y-2|$. Since the minimum value for the difference between the y-coordinates is at $y=0$, we get $2x+4=20$ or $-2x+4=20$. Thus, the upper and lower bounds for $x$ are $8$ and $-8$, respectively. Now we test each value for x satisfying $3<x\le 8$ and double the result because of symmetry. For $x=4$, the possibles values of y are such that $|2y|\le 12$ for a total of $13$ lattice points, for $x=5$, the possibles values of y are such that $|2y|\le 10$ for a total of $11$ lattice points, for $x=6$, the possibles values of y are such that $|2y|\le 8$ for a total of $9$ lattice points, for $x=7$, the possibles values of y are such that $|2y|\le 6$ for a total of $7$ lattice points, for $x=8$, the possibles values of y are such that $|2y|\le 4$ for a total of $5$ lattice points, Hence, there are a total of $105+2(13+11+9+7+5)=\boxed{195}$ lattice points. One may also obtain the result by using Pick's Theorem(how?). $i=a-\frac{b}{2}-1$ (Suggestion)