# A different Traveling Salesman problem [closed]

A salesman travelled due west from city A to city B. The distance he travelled was X km. He returned from B to A and found that he had travelled half the distance i.e. 1⁄2X. How can that be?

## closed as too broad by boboquack, Jamal Senjaya, ABcDexter, Deusovi♦Sep 15 '17 at 15:22

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• Are the cities considered "points" for this puzzle? Can half of the first trip be done while still staying within the city limits of City A? – Keeta Sep 14 '17 at 13:37
• This seems far too broad to have one single answer, apart from a "guess the number I'm thinking of" right answer according to @amirul – Matt Taylor Sep 14 '17 at 13:50
• Edges in TSP can have different weights. Or is this a planar TSP (one where distance is calculated by absolute difference of coordinates on 2D plane? – rus9384 Sep 14 '17 at 19:41

Maybe

A is two thirds of the way around the globe

so

to return to B, the shortest path would be to continue the last third -- 1/2X

There are probably

Lots of other tricks with a globe, e.g the salesman is near the north pole, for example.

• Won't' exactly be 2/3 since the rotation of earth should also be taken into account. – Laschet Jain Sep 14 '17 at 8:25
• No rotation would not have effect on the travel speed. – Burak Mete Sep 14 '17 at 8:27
• @BurakMete Well, since he travelled due west? So this is plausible – Wen1now Sep 14 '17 at 8:46
• Right, makes sense – Burak Mete Sep 14 '17 at 8:47
• @LashitJain the question is about distance, not time, so the earth's rotation wouldn't affect the answer. – Herb Wolfe Sep 14 '17 at 11:31

Simple:

A and B are the same city. X=0km.

• come on!! ..... – sousben Sep 14 '17 at 9:26
• or there at at an infinite distance from each other. – Marius Sep 14 '17 at 11:12
• If x = 0, then x is a point and thus has no direction (as direction is determined by a change in position), so he couldn't have traveled due west in this scenario. – maxathousand Sep 14 '17 at 15:22
• This is good lateral thinking! – Brian Risk Sep 14 '17 at 18:49
• Oh Jon, we need a non-trivial solution to this problem! – ABcDexter Sep 15 '17 at 15:17

Both City A and City B are on the equator, City B was 28000km west of A, he went back going west as well, which would be 14000km.

• Your answer is correct given the information but the OP should give time to travel too. Since the earth rotates and we travel relative to it. – Laschet Jain Sep 14 '17 at 8:23
• @LashitJain No no no, that would overcomplicate the question. We assume the earth is stationary - due to relativity, there is no 'absolute frame or reference' – Wen1now Sep 14 '17 at 8:47
• It doesn't have to be on the equator – somebody Sep 14 '17 at 11:29
• @Wen1now there's no absolute frame of reference, but there is the CMB stationary frame of reference (600 km/s relative to Earth, so not relevant here) and there's the inertial frame of reference comoving with Earth. But then again "due west" is only defined in frames of reference in which the surface of Earth is relatively stationary. – John Dvorak Sep 14 '17 at 13:12
• @somebody with the distances i stated, it needs to be :P – DrunkWolf Sep 15 '17 at 6:20

He traveled through mountain curvy roads, seasides etc...

and after that

he turned back by plane by (almost) straight line

• Well, then he wouldn't be travelling due west – boboquack Sep 14 '17 at 10:08
• Hmm, I considered word 'due west' only to be applicable to the first sentence... – mpasko256 Sep 14 '17 at 12:40
• @mpasko256 The first sentence is the travel from A to B. You described that part as "He traveled through mountain curvy roads, seasides etc..." – JiK Sep 14 '17 at 13:46
• Using a similar logic then maybe they were launched out of a cannon or went over a mountain such that their path was roughly an arc or curve with a length of X but the path on the ground was 1/2X. This would allow the cardinal direction to be due West even though the actual path goes "around". Clever answer! – Lunin Sep 14 '17 at 20:42
• @boboquack Can't you travel 'due west' in a zigzag? With how the question is formulated this is a good answer. – Oleg Sep 17 '17 at 5:26

While he travaled due west to get there, while returning, he

followed the Geodesic.

Although I am not sure if that can make up for 50%.

• If you're only allowed to travel less than 180 degrees in longitude, even close to a pole, the ratio between distances between a constant latitude path and a geodesic path is at most $\pi/2$. But the first path is allowed to go over 180 degrees in longitude, then it's clearly possible in any latitude (even in equator, see other answers). – JiK Sep 14 '17 at 13:44

At least one of the cities could have moved.

This could be easily true with a 'tent city' or cities on different planets. For a space trip 'due west' could be a course tangential to the surface.

• If a city moved do you still 'return' to it? – Oleg Sep 17 '17 at 5:30

He went straight over a mountain the first time, and though it's tunnel on the way back.

The frame of reference for measuring the distances is the Solar System. Taking into account rotation of the Earth and its orbit of the Sun, journey A-B started and concluded at fixed points in the Solar System that were twice the straight-line separation of those of journey B-A; because journey A-B took roughly twice the duration.

Maybe

City A is on a hill. He traveled a longer route going downhill because going down to steeply is harder/too dangerous, but on the way back he just took the most direct route.

• A small problem - he wouldn't have travelled due west. By the way, welcome to Puzzling! Why don't you take the tour? – boboquack Sep 15 '17 at 2:39

This saleman was

on a boat and had to go against the river to go, and let the water do the job on returning?

He HAD in fact travelled half the distance (1/2X). Then he had travelled half the distance again (12/X), and got back to where he started (After a total distance of X).

You could even read the story as being told out of order, if that makes more sense.

• @maxathousand "He returned from B to A and found that he had travelled half the distance" Nowhere does it say he only travelled that distance, and no more than that. He had indeed travelled that. Twice. – Weckar E. Sep 14 '17 at 15:38
• That's the same as saying that you hold 3 apples while you are holding 6 of them. – Hans Janssen Sep 14 '17 at 15:41
• @WeckarE. So by that logic, all of my friends are 2 years old. – maxathousand Sep 14 '17 at 15:59
• @Geliormth Technically you are. Lateral thinking, right? – Weckar E. Sep 14 '17 at 16:34
• @maxathousand No, but they have lived for two years. – Weckar E. Sep 14 '17 at 16:35