Given an arrangement of blobs, how can you determine whether it is possible to exterminate them all?
First, let's color the grid in a checkerboard pattern. We'll call the groups of squares light squares and dark squares. Note the effect of each possible move on the count of blobs on each group:
• provoking: -1 light, -1 dark
• poking a light square: -1 light, +4 dark
• poking a dark square: +4 light, -1 dark
Observe that in each case, the difference between the number of blobs on light and dark squares changes by a multiple of 5 (0 or ±5). Therefore, given an arrangement of blobs, if the difference between the number of blobs on light and dark squares is not a multiple of 5, then extermination is impossible. All that remains is to show the converse: Extermination is possible if the difference is a multiple of 5.
What strategy can you use to succeed when possible?
Warning: what follows is a constructive but highly inefficient strategy. Do not try this at home.
- Exterminate blobs on light squares.
- Show how to move a blob orthogonally.
- Show how to move a solitary blob diagonally.
- Show how to move a crowded blob diagonally.
- Move blobs to where they can be exterminated in groups.
Light square blobs
While there exists some blob on a light square, exterminate it by either provoking it with a neighboring blob on a dark square or poking it if it has no neighbors. From now on, we'll assume that all blobs are on dark squares except during intermediate steps.
If the circled squares are empty, we can move this blob 2 spaces to the right like this:
Isolated diagonal movement
If the circled squares are empty, we can move this blob diagonally 1 space up and right like this:
Crowded diagonal movement
With extra work, we can move a blob diagonally even with blobs on the circled squares above as long as the destination square is free. Call the blob to move Bob.
- First, let's give Bob some breathing room. If there are any blobs in columns to the right of Bob, then take the blobs in the column farthest to the right and move them 2 spaces farther right, one at a time. Repeat this with the second farthest column with blobs, if it exists, etc. Repeat all of this in each of the other 3 directions.
- After clearing the dance floor, watch Bob perform the electric slide as before.
- Finally, perform the same moves in step 1 in reverse order and direction to restore everyone other than Bob to their original squares.
Pick an empty group of 5 dark squares far away from any blobs, in the pattern shown above. This will be the designated extermination zone. While there are at least 5 blobs, move the 5 closest to the zone to it, one at a time, using a sequence of diagonal moves. Then exterminate them like so:
Repeat until fewer than 5 blobs remain.
If the difference between the number of blobs on light and dark squares at the start was a multiple of 5, then it still is. Because there are no blobs on light squares and fewer than 5 blobs on dark squares, the number of blobs on dark squares must be 0. Extermination complete!