Around the world - Revision history

Mrd000: Undo revision 1396 by Mrd000 (talk) (realised how it works)

2021-01-24T21:40:40Z

Undo revision 1396 by Mrd000 (talk) (realised how it works)

← Older revision		Revision as of 15:40, 24 January 2021
Line 80:		Line 80:
	---		---

	Here is a solution involving only three planes, including the one around the world (call this one the "main plane"). Two planes (Plane 1 and Plane 2) escort the main plane, all fully fueled. At 1/8 of the way around the world, Plane 1 fills up Plane 2 and the main plane, then returns to the island. At 1/4 of the way, Plane 2 fills up the main plane and returns to the island. At half way around, Plane 1, fully fueled, takes off again, now in the opposite direction as previously. When it meets the main plane, which is now 3/4 of the way around the world, Plane 1 gives the main plane 1/4 of a tank. They both fly toward the island, and a fully fueled Plane 2 leaves the island, flying toward them. Halfway through their return (so the main plane is now 7/8 the way around the world), Plane 2 meets them and gives each 1/4 of a tank. All three planes can now safely return to the island.		Here is a solution involving only three planes, including the one around the world (call this one the "main plane"). Two planes (Plane 1 and Plane 2) escort the main plane, all fully fueled. At 1/8 of the way around the world, Plane 1 fills up Plane 2 and the main plane, then returns to the island. At 1/4 of the way, Plane 2 fills up the main plane and returns to the island. At half way around, Plane 1, fully fueled, takes off again, now in the opposite direction as previously. When it meets the main plane, which is now 3/4 of the way around the world, Plane 1 gives the main plane 1/4 of a tank. They both fly toward the island, and a fully fueled Plane 2 leaves the island, flying toward them. Halfway through their return (so the main plane is now 7/8 the way around the world), Plane 2 meets them and gives each 1/4 of a tank. All three planes can now safely return to the island.}}

	Does this last solution actually work? When Plane 2 fills the Main Plain up at 1/4 it will have no enough fuel left to return (it spent 1/4 flying and 1/8 of fuel to the Main Plane, so it only has 1/8 to return 1/4 way).
	}}

	[[Category: Optimization puzzles]]		[[Category: Optimization puzzles]]
	[[Category: Geometry]]		[[Category: Geometry]]

Mrd000: Undo revision 1397 by Mrd000 (talk)

2021-01-24T21:40:15Z

Undo revision 1397 by Mrd000 (talk)

← Older revision		Revision as of 15:40, 24 January 2021
Line 82:		Line 82:
	Here is a solution involving only three planes, including the one around the world (call this one the "main plane"). Two planes (Plane 1 and Plane 2) escort the main plane, all fully fueled. At 1/8 of the way around the world, Plane 1 fills up Plane 2 and the main plane, then returns to the island. At 1/4 of the way, Plane 2 fills up the main plane and returns to the island. At half way around, Plane 1, fully fueled, takes off again, now in the opposite direction as previously. When it meets the main plane, which is now 3/4 of the way around the world, Plane 1 gives the main plane 1/4 of a tank. They both fly toward the island, and a fully fueled Plane 2 leaves the island, flying toward them. Halfway through their return (so the main plane is now 7/8 the way around the world), Plane 2 meets them and gives each 1/4 of a tank. All three planes can now safely return to the island.		Here is a solution involving only three planes, including the one around the world (call this one the "main plane"). Two planes (Plane 1 and Plane 2) escort the main plane, all fully fueled. At 1/8 of the way around the world, Plane 1 fills up Plane 2 and the main plane, then returns to the island. At 1/4 of the way, Plane 2 fills up the main plane and returns to the island. At half way around, Plane 1, fully fueled, takes off again, now in the opposite direction as previously. When it meets the main plane, which is now 3/4 of the way around the world, Plane 1 gives the main plane 1/4 of a tank. They both fly toward the island, and a fully fueled Plane 2 leaves the island, flying toward them. Halfway through their return (so the main plane is now 7/8 the way around the world), Plane 2 meets them and gives each 1/4 of a tank. All three planes can now safely return to the island.

	~~[[User:mrd000]]:~~ Does this last solution actually work? When Plane 2 fills the Main Plain up at 1/4 it will have no enough fuel left to return (it spent 1/4 flying and 1/8 of fuel to the Main Plane, so it only has 1/8 to return 1/4 way).		Does this last solution actually work? When Plane 2 fills the Main Plain up at 1/4 it will have no enough fuel left to return (it spent 1/4 flying and 1/8 of fuel to the Main Plane, so it only has 1/8 to return 1/4 way).
	}}		}}

	[[Category: Optimization puzzles]]		[[Category: Optimization puzzles]]
	[[Category: Geometry]]		[[Category: Geometry]]

Mrd000: added who wrote the last comment

2021-01-24T20:51:46Z

added who wrote the last comment

← Older revision		Revision as of 14:51, 24 January 2021
Line 82:		Line 82:
	Here is a solution involving only three planes, including the one around the world (call this one the "main plane"). Two planes (Plane 1 and Plane 2) escort the main plane, all fully fueled. At 1/8 of the way around the world, Plane 1 fills up Plane 2 and the main plane, then returns to the island. At 1/4 of the way, Plane 2 fills up the main plane and returns to the island. At half way around, Plane 1, fully fueled, takes off again, now in the opposite direction as previously. When it meets the main plane, which is now 3/4 of the way around the world, Plane 1 gives the main plane 1/4 of a tank. They both fly toward the island, and a fully fueled Plane 2 leaves the island, flying toward them. Halfway through their return (so the main plane is now 7/8 the way around the world), Plane 2 meets them and gives each 1/4 of a tank. All three planes can now safely return to the island.		Here is a solution involving only three planes, including the one around the world (call this one the "main plane"). Two planes (Plane 1 and Plane 2) escort the main plane, all fully fueled. At 1/8 of the way around the world, Plane 1 fills up Plane 2 and the main plane, then returns to the island. At 1/4 of the way, Plane 2 fills up the main plane and returns to the island. At half way around, Plane 1, fully fueled, takes off again, now in the opposite direction as previously. When it meets the main plane, which is now 3/4 of the way around the world, Plane 1 gives the main plane 1/4 of a tank. They both fly toward the island, and a fully fueled Plane 2 leaves the island, flying toward them. Halfway through their return (so the main plane is now 7/8 the way around the world), Plane 2 meets them and gives each 1/4 of a tank. All three planes can now safely return to the island.

	Does this last solution actually work? When Plane 2 fills the Main Plain up at 1/4 it will have no enough fuel left to return (it spent 1/4 flying and 1/8 of fuel to the Main Plane, so it only has 1/8 to return 1/4 way).		[[User:mrd000]]: Does this last solution actually work? When Plane 2 fills the Main Plain up at 1/4 it will have no enough fuel left to return (it spent 1/4 flying and 1/8 of fuel to the Main Plane, so it only has 1/8 to return 1/4 way).
	}}		}}

	[[Category: Optimization puzzles]]		[[Category: Optimization puzzles]]
	[[Category: Geometry]]		[[Category: Geometry]]

Mrd000: don't think the three plane solution works

2021-01-24T20:49:44Z

don't think the three plane solution works

← Older revision		Revision as of 14:49, 24 January 2021
Line 80:		Line 80:
	---		---

	Here is a solution involving only three planes, including the one around the world (call this one the "main plane"). Two planes (Plane 1 and Plane 2) escort the main plane, all fully fueled. At 1/8 of the way around the world, Plane 1 fills up Plane 2 and the main plane, then returns to the island. At 1/4 of the way, Plane 2 fills up the main plane and returns to the island. At half way around, Plane 1, fully fueled, takes off again, now in the opposite direction as previously. When it meets the main plane, which is now 3/4 of the way around the world, Plane 1 gives the main plane 1/4 of a tank. They both fly toward the island, and a fully fueled Plane 2 leaves the island, flying toward them. Halfway through their return (so the main plane is now 7/8 the way around the world), Plane 2 meets them and gives each 1/4 of a tank. All three planes can now safely return to the island.}}		Here is a solution involving only three planes, including the one around the world (call this one the "main plane"). Two planes (Plane 1 and Plane 2) escort the main plane, all fully fueled. At 1/8 of the way around the world, Plane 1 fills up Plane 2 and the main plane, then returns to the island. At 1/4 of the way, Plane 2 fills up the main plane and returns to the island. At half way around, Plane 1, fully fueled, takes off again, now in the opposite direction as previously. When it meets the main plane, which is now 3/4 of the way around the world, Plane 1 gives the main plane 1/4 of a tank. They both fly toward the island, and a fully fueled Plane 2 leaves the island, flying toward them. Halfway through their return (so the main plane is now 7/8 the way around the world), Plane 2 meets them and gives each 1/4 of a tank. All three planes can now safely return to the island.

			Does this last solution actually work? When Plane 2 fills the Main Plain up at 1/4 it will have no enough fuel left to return (it spent 1/4 flying and 1/8 of fuel to the Main Plane, so it only has 1/8 to return 1/4 way).
			}}

	[[Category: Optimization puzzles]]		[[Category: Optimization puzzles]]
	[[Category: Geometry]]		[[Category: Geometry]]

Oscarlevin at 04:30, 8 July 2013

2013-07-08T04:30:26Z

← Older revision		Revision as of 22:30, 7 July 2013
Line 1:		Line 1:
			[[File:Airplane Silhouette R.svg\|right\|150px]]

	==Puzzle==		==Puzzle==

Urlooney at 20:47, 13 June 2012

2012-06-13T20:47:00Z

← Older revision		Revision as of 14:47, 13 June 2012
Line 1:		Line 1:
			==Puzzle==

	A group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. Any desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island, and it is assumed that there is no time lost in refueling either in the air or on the ground. What is the smallest number of planes that will ensure the flight of one plane around the world on a great circle, assuming that the planes have the same constant speed (relative to the ground) and rate of fuel consumption, and that all planes return safely to their island base?		A group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. Any desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island, and it is assumed that there is no time lost in refueling either in the air or on the ground. What is the smallest number of planes that will ensure the flight of one plane around the world on a great circle, assuming that the planes have the same constant speed (relative to the ground) and rate of fuel consumption, and that all planes return safely to their island base?

Urlooney at 20:45, 13 June 2012

2012-06-13T20:45:35Z

← Older revision		Revision as of 14:45, 13 June 2012
Line 72:		Line 72:
	This involves a total of six flights: one main, five returning. But by the time the eastern assisting flights have to depart, the western assisting flights have returned to the island. So the same aeroplanes can be re-used for the eastern assisting flights, reducing the number of aeroplanes needed by two.		This involves a total of six flights: one main, five returning. But by the time the eastern assisting flights have to depart, the western assisting flights have returned to the island. So the same aeroplanes can be re-used for the eastern assisting flights, reducing the number of aeroplanes needed by two.

	Therefore the puzzle can be solved with a total of four aeroplanes, including the main one. But certainly no fewer. Nothing is wasted in the above scenario, which confirms that it is the optimal solution. }}		Therefore the puzzle can be solved with a total of four aeroplanes, including the main one. But certainly no fewer. Nothing is wasted in the above scenario, which confirms that it is the optimal solution.

			---

			Here is a solution involving only three planes, including the one around the world (call this one the "main plane"). Two planes (Plane 1 and Plane 2) escort the main plane, all fully fueled. At 1/8 of the way around the world, Plane 1 fills up Plane 2 and the main plane, then returns to the island. At 1/4 of the way, Plane 2 fills up the main plane and returns to the island. At half way around, Plane 1, fully fueled, takes off again, now in the opposite direction as previously. When it meets the main plane, which is now 3/4 of the way around the world, Plane 1 gives the main plane 1/4 of a tank. They both fly toward the island, and a fully fueled Plane 2 leaves the island, flying toward them. Halfway through their return (so the main plane is now 7/8 the way around the world), Plane 2 meets them and gives each 1/4 of a tank. All three planes can now safely return to the island.}}

	[[Category: Optimization puzzles]]		[[Category: Optimization puzzles]]
	[[Category: Geometry]]		[[Category: Geometry]]

Oscarlevin: Reverted edits by Ocafaromy (talk) to last revision by Zerrakhi

2010-11-24T12:52:13Z

Reverted edits by Ocafaromy (talk) to last revision by Zerrakhi

Show changes

Ocafaromy at 23:45, 23 November 2010

2010-11-23T23:45:09Z

Show changes

Zerrakhi at 07:25, 25 October 2010

2010-10-25T07:25:02Z

← Older revision		Revision as of 01:25, 25 October 2010
Line 1:		Line 1:
	A group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. Any desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island, and it is assumed that there is no time lost in refueling either in the air or on the ground. What is the smallest number of planes that will ensure the flight of one plane around the world on a great circle, assuming that the planes have the same constant speed (relative to the ground) and rate of fuel consumption, and that all planes return safely to their island base?		A group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. Any desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island, and it is assumed that there is no time lost in refueling either in the air or on the ground. What is the smallest number of planes that will ensure the flight of one plane around the world on a great circle, assuming that the planes have the same constant speed (relative to the ground) and rate of fuel consumption, and that all planes return safely to their island base?

	{{Solution \| First, let's agree on conventions. The main journey will be westward, and all units will be in degrees. So 360 degrees is a full circle around the earth, 1 unit of fuel is enough fuel to travel one degree, and 1 unit of time is the time taken to travel one degree.		{{Solution \| [Update: I (User:Zerrakhi) have since learned that the following solution is wrong. There is a better and simpler solution that uses only three aircraft and relies on a step that I missed. But I don't have the heart to delete all my hard work.]

			First, let's agree on conventions. The main journey will be westward, and all units will be in degrees. So 360 degrees is a full circle around the earth, 1 unit of fuel is enough fuel to travel one degree, and 1 unit of time is the time taken to travel one degree.

	One unassisted aeroplane can travel 180 degrees non-returning, or 90 degrees returning. Let's see how far we can get with two aeroplanes: a main flight and an assisting flight.		One unassisted aeroplane can travel 180 degrees non-returning, or 90 degrees returning. Let's see how far we can get with two aeroplanes: a main flight and an assisting flight.