Problem 165 Answers

1. At time t=0 light fuse 1 at both ends and fuses 2, 3, 4, and 5 at one end. Fuse 1 will have 32 minutes remaining, fuses 2-5 will have 64 minutes remaining.
2. At time t=32 fuse 1 will have burned out. At this moment light fuse 2 at the other end. Fuse 2 will have 16 minutes remaining, fuses 3-5 will have 32 minutes remaining.
3. At time t=48 fuse 2 will have burned out. At this moment light fuse 3 at the other end. Fuse 3 will have 8 minutes remaining, fuses 4,5 will have 16 minutes remaining.
4. At time t=56 fuse 3 will have burned out. At this moment light fuse 4 at the other end. Fuse 4 will have 4 minutes remaining, fuse 5 will have 8 minutes remaining.
5. At time t=60 fuse 4 will have burned out. At this moment start fuse 6. Fuse 5 will have 4 minutes remaining, fuse 6 will have 64 minutes remaining.
6. At time t=64 fuse 5 will have burned out. At this moment snuff out fuse 6. Fuse 6 will have 60 minutes remaining when relit.

Next is another solution submitted by Michael Nolan.

Solution 2

1. At time t=0 light fuse 1 at both ends and fuse 2 at one end. Fuse 1 will have 32 minutes remaining and fuse 2 will have 64 minutes remaining.
2. At time t=32 fuse 1 will have burned out. At this moment light the other end of fuse 2 and fuse 3 at one end. Fuse 2 will have 16 minutes remaining and fuse 3 will have 64 minutes remaining.
3. At time t=48 fuse 2 will have burned out. At this moment light fuse 3 at the other end and light fuse 4 at one end. Fuse 3 will have 24 minutes remaining and fuse 4 will have 64 minutes remaining.
4. At time t=72 fuse 3 will have burned out. At this moment light fuse 5 from both ends. Fuse 4 will have 40 minutes remaining and fuse 5 will have 32 minutes remaining.
5. At time t=104 fuse 5 will have burned out. At this moment light fuse 4 from the other end and light fuse 6 from one end. Fuse 4 will have 4 minutes remaining and fuse 6 will have 64 minutes remaining.
6. At time t=108 fuse 4 will have burned out. At this moment snuff out fuse 6. Fuse 6 will have 60 minutes remaining.

Next is another solution submitted by Michael Nolan and Andy 'Deuce' Smith.

Solution 3

1. Light #1 at both ends, and #2 and #3 at one end. Unlight #2 and #3 when #1 burns out. This makes #2 and #3 both 32-minute fuses.
2. Light #2 at both ends, and #4 at one end. Unlight #4 when #2 burns out. This makes #4 a 48-minute fuse.
3. Light #4 at both ends, and #3 at one end. Unlight #3 when #4 burns out. This makes #3 an 8-minute fuse.
4. Light #3 at both ends, and #5 at one end. Unlight #5 when #3 burns out. This makes #5 a 60-minute fuse.

Solution 4

1. At time t=0 light fuse 1 at both ends and fuses 2, 3, and 4 at both ends. Fuse 1 will have 32 minutes remaining, fuses 2-4 will have 64 minutes remaining.
2. At time t=32 fuse 1 will have burned out. At this moment light fuse 2 at the other end. Fuse 2 will have 16 minutes remaining, fuses 3,4 will have 32 minutes remaining.
3. At time t=48 fuse 2 will have burned out. At this moment light fuse 3 at the other end. Fuse 3 will have 8 minutes remaining, fuses 4 will have 16 minutes remaining.
4. At time t=56 fuse 3 will have burned out. At this moment light fuse 4 at the other end and fuse 5 at one end. Fuse 4 will have 4 minutes remaining and fuse 5 will have 64 minutes remaining.
5. At time t=60 fuse 4 will have burned out. At this moment snuff out fuse 5. Fuse 5 will have 60 minutes remaining when relit.

Credit and thanks for this problem goes to Craig Olson, Alan Goldberg, and the Nov/Dec 1999 Contingencies magazine.