Staying close to home

As a child, I used to puzzle over a self-made conundrum, which is similar to the taxi problem I set a while ago. Here it is.

Imagine the eight lines that connect the 16 primary points of the compass (N, NNE, NE, ENE, E, ESE, SE, SSE, S, SSW, SW, WSW, W, WNW, NW and NNW). Each line connects two of these sixteen points. So, for instance, the line from N to S constitutes a single line, as does that connecting SSE and NNW.

Each line has the same length: one kilometre, let’s say.

You walk each of the lines in succession, but for each line, you can choose which direction to take. So, for the N/S line, you can either walk north or south for one kilometre. After doing so, you take the NNE/SSW line from your previous end point, again in either direction. Etc.

Here’s the conundrum: is it possible to end up where you started. And if not, how close to the start can you end up and how do you do this?

I’ve just worked out the answer, which I’ll post as a comment.

Comments

One Response to “Staying close to home”

  1. Dan on December 25th, 2006 04:42

    It’s not possible to end up where you started. The closest you can end up is 0.422km from home.

    There are 256 options available to you, as you have eight binary choices. 16 of these will leave you 0.422km from home. (A further 16 will leave you 5.13km from home.)

    The 16 journeys which will leave you closest to home are:

    N, SSW, NE, ENE, W, WNW, SE, SSE
    S, NNE, SW, WSW, E, ESE, NW, NNW
    N, SSW, SW, ENE, E, WNW, NW, SSE
    S, NNE, NE, WSW, W, ESE, SE, NNW
    N, NNE, SW, ENE, W, WNW, SE, SSE
    N, NNE, SW, WSW, E, ESE, NW, SSE
    N, SSW, SW, ENE, E, WNW, SE, NNW
    S, NNE, NE, WSW, W, ESE, NW, SSE
    S, SSW, NE, ENE, W, WNW, SE, NNW
    S, SSW, NE, WSW, E, ESE, NW, NNW
    N, SSW, NE, WSW, W, ESE, SE, NNW
    N, SSW, SW, ENE, W, ESE, SE, NNW
    S, NNE, NE, WSW, E, WNW, NW, SSE
    S, NNE, SW, ENE, E, WNW, NW, SSE
    N, NNE, SW, WSW, E, WNW, SE, SSE
    S, SSW, NE, ENE, W, ESE, NW, NNW

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