Possibility question; isn't the answer 2?

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Discussion Overview

The discussion revolves around a mathematical problem involving a boy walking in two dimensions, specifically focusing on the number of distinct paths he can take to reach a point 8 kilometers north and 8 kilometers east from his starting position. The conversation explores interpretations of the problem, particularly regarding restrictions on movement direction.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests the answer is 2, based on two possible starting directions (east or north), questioning the complexity of the problem.
  • Another participant agrees that the question is poorly worded and asks for clarification on what the answer pertains to.
  • Some participants argue that for the answer to be 256, the restriction on not walking in the same direction consecutively must be removed.
  • One participant proposes an alternative interpretation that allows for a maximum deviation of 1 kilometer from a direct diagonal path, which could also lead to the answer being 256.
  • A later reply reflects on the nature of the discussion, suggesting that exploring the nuances of the question may be seen as unusual but enjoyable.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the problem and the conditions under which the answer changes. There is no consensus on the correct interpretation or final answer.

Contextual Notes

Participants note the importance of the restrictions on movement direction and how they affect the calculation of possible paths. The discussion highlights the ambiguity in the problem's wording and the assumptions made by different participants.

x86
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So I'm doing some math stuff on this internet website for fun, aplusclick. Anyway, there was this question on the site:

"A boy begins walking from his starting point. Each hour, he either walks one kilometer to the east or one kilometer to the north, but he never walks in the same direction.

In how many different ways can he get to a point that is 8 kilometers to the north and 8 kilometers to the east of his starting point?"

The answer is 2^8, or 256. But shouldn't it be 2?

He can either start by going east, then north, then east - until his displacement is 8 km N, 8 km E
or he can start by going north, then east, then north, etc

So shouldn't it be 2? Can someone break this down for me if I am wrong?
 
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You are correct - the question is poorly worded: what question does the answer belong to?
 
Yeah, for the answer to be 256, you have to remove the restriction that he never walks the same way for two consecutive 1KM legs.
 
phinds said:
Yeah, for the answer to be 256, you have to remove the restriction that he never walks the same way for two consecutive 1KM legs.

But then the answer is 16!/(8!8!) =12870
 
To get 256, he has to walk for 8 hours, without a restriction on walking directions.

An alternative interpretation: He never deviates more than 1km north or east from the direct (diagonal) path, so the difference between "hours moved north" and "hours moved east" is at least -1 and at most 1. That would give 256 as well.
 
You know, most people would have responded to the question as x86 did and leave it at that.
I sometimes wonder if we, who go explore things like which question the answer belongs to are just plain weird.
Mind you - it more fun this way :)
 

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