# The Origins of a Confusing Maze

Behold a maze of fuses (the black dotted lines) laid out on a grid:

The fuses can be lit, causing sparks to burn along them at a perfectly uniform rate: one segment per second. Your goal is to light the fuses in exactly four (4) places so that all sparks extinguish at exactly the same time.

The rules:

1. fuses may only be lit on the corners of grid squares (that is, the "gaps" between the fuse segments); all fuses must be lit at the same time
2. if $n$ spark(s) meet at an $n$-way intersection, they extinguish; this includes the case of a single spark reaching a dead-end
3. if $n$ spark(s) meet at a $k$-way intersection, with $k > n$, they do not extinguish; instead, they split or merge if necessary and continue to follow all outgoing branches (all branches that sparks didn't come in on)

Can you figure out which four places to light the fuses?

Hint: See https://i.sstatic.net/lu4qO.png

The four colours (red, purple, blue, and green) correspond to fuses burned by sparks spreading from the first, second, third, and fourth spark origins, respectively.

## Example

As an example, consider one part of the grid:

Lighting the fuse at D5 is a potential solution. It will start burning north and south, reaching B3 in 4s and B6 at 3s, splitting at both. The B3 north spark and B6 south spark will continue burning (exiting the grid). The B3 south and B6 north sparks will reach B4 at exactly the same time (5s) and merge, continuing to burn along the western branch.

Lighting the fuse at D4 is not a potential solution. We see that in this case, two sparks will meet and extinguish at B5 after 5s, but other sparks (at A2 and A4, and possibly B7) will still be burning, meaning all sparks do not extinguish at the same time.

• I'm a little confused by how the extinguishing happens. What would happen next for this? The green circle represents the 4 places that were lit. Commented Jun 23, 2015 at 4:52
• @MikeEarnest: Thanks. I've made the fix.
– COTO
Commented Jun 23, 2015 at 8:09
• @Allan: The four fuses meeting at the four way intersection would extinguish per rule 2. The four fuses not at the intersection would all still be burning at this point, violating the "all fuses must extinguish at the same time" requirement and disqualifying this as a solution.
– COTO
Commented Jun 23, 2015 at 8:14
• @IvoBeckers: I came up with the concept and this puzzle. It may well have been done before, but I don't know where or by whom. I'll put up a slightly different take on the genre later this week. I think this one was a bit too difficult for most.
– COTO
Commented Jun 24, 2015 at 18:59
• I know it's solved already, but it was fun... I built an interactive animation for this: check it out
– JNF
Commented Jun 29, 2015 at 12:32

I think the below picture is your intended solution where red dots are where the fuses are lit and black where they die out. however, the error is on the pink dot because at that point 2 sparks need to merge but they arive there on different times. Nice puzzle though! Kept me quite busy :)

• That's excellent. It's the correct solution. The error is a "transcription error" of sorts from the model I published vs. the model I used for validation. I'll award you 50 rep when the option becomes available for not only solving the puzzle, but fixing it as well. Great work. ;)
– COTO
Commented Jun 24, 2015 at 3:49
• I've amended the puzzle and the hint to remedy the error.
– COTO
Commented Jun 24, 2015 at 3:57
• +1 great job. What technic did you use to solve that?
– A.D.
Commented Jun 24, 2015 at 15:37
• Well. At first I used the fact that the dead ends should always be start or end points. Trying them first as starting points I found out quickly that the top two ones at least couldn't be starting points
– Ivo
Commented Jun 24, 2015 at 18:25
• @IvoBeckers, how is this for a test ground?
– JNF
Commented Jun 29, 2015 at 21:15