A283075 Number of time intervals that can be measured with n ropes and a lighter from the moment when the first rope begins burning.

2, 5, 11, 23, 48, 101, 218, 473, 1044, 2284, 5011, 10979, 23954

Offset: 1

Author

Samuel Cormier-Iijima, Feb 28 2017

Keywords

• nonn
• hard
• more

Comments

Suppose you have n ropes and a lighter. Each rope burns at a nonconstant rate but takes exactly one hour to burn completely from one end to the other. You can only light the ropes at either of their ends but can decide when to light each end as you see fit. If you're strategic in how you burn the ropes, how many specific lengths of time can you measure? (For example, if you had one rope, you could measure two lengths of time: one hour, by simply burning the entire rope from one end, and half an hour, by burning the rope from both ends and marking when the flames meet.)

In this sequence, the time intervals begin when the first rope (or safety fuse, or match cord) begins burning. Without this restriction, if we are allowed to burn some ropes beforehand, more time intervals are available for measuring off. - Andrey Zabolotskiy, Feb 28 2017

Examples

			a(2)=5: (i) Generate 1 by burning one rope from one end. (ii) Generate 2 by burning one rope from one end at t=0 and the other afterwards at t=1 from one end. (iii) Generate 1/2 by burning 1 rope from both ends. (iv) Generate 3/2 by burning 1 rope from one end at t=0 then the other from both ends at t=1 (or swapped order). (v) Generate 3/4 by burning one rope at t=0 from both ends, starting the other also at t=0 at one end, and lighting the other's second end at t=1/2 when the first rope's flames have reached the middle, so the 2nd rope's flames finish at t=3/4.
For n = 3 the a(3) = 11 solutions are 1/2, 3/4, 7/8, 1, 9/8, 5/4, 3/2, 7/4, 2, 5/2, 3.
		

Links

• FiveThirtyEight, The Riddler (see section Riddler Classic), see also the solution (for a(4))
• Samuel Cormier-Iijima, Python implementation

Crossrefs

Cf. A287012.

Extensions

a(9) from Andrey Zabolotskiy, Mar 02 2017

a(10)-a(13) from Mark Rickert, Apr 10 2017

Name clarified by Mark Rickert, Jun 08 2017