1. R
    Standard memberRemoved
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    06 Mar '21 14:113 edits
    @venda said
    I've tried all that,with little dots on a piece of paper.
    First, I went up to 7 "dots" in a straight line east and put a dot at north and south at every step.
    I then added up the dots and came up with the answer 3k+1(1 representing an 8th step east)
    I then tried with random directions (e.g 2 steps east 1 step north etc) but could only count 3k this way.
    I suspect I need a different approach or a bigger piece of paper!!
    If you want the answer let me know...I'm not keeping it from you, but if you want to work it out, lets start with this:

    For k = 1 the graph should look like this:

    .....♥
    .♥ ♦ ♥
    .....♥

    With 1 step

    The ♥ are the spots you can get to, the ♦ are spots you can't get to.

    N_s = 4


    ( You don't have to make a picture, but if you wish to this is where I got the symbols - https://www.alt-codes.net/ just click them and they copy, then just paste them )

    For k= 2 ?
  2. Subscribervenda
    Dave
    S.Yorks.England
    Joined
    18 Apr '10
    Moves
    83457
    06 Mar '21 16:48
    @joe-shmo said
    If you want the answer let me know...I'm not keeping it from you, but if you want to work it out, lets start with this:

    For k = 1 the graph should look like this:

    .....♥
    .♥ ♦ ♥
    .....♥

    With 1 step

    The ♥ are the spots you can get to, the ♦ are spots you can't get to.

    N_s = 4


    ( You don't have to make a picture, but if you wish to this is where I got the ...[text shortened]... - https://www.alt-codes.net/ just click them and they copy, then just paste them )

    For k= 2 ?
    I can see where this is going I think.
    I'll give it one more look later.
    You may be interested in the puzzle I've put up on the newspaper thread.
    I can't see a way to do it mathematicallyThread 188467
  3. Joined
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    07 Mar '21 10:54
    100%
  4. Subscribervenda
    Dave
    S.Yorks.England
    Joined
    18 Apr '10
    Moves
    83457
    09 Mar '21 19:21
    @joe-shmo said
    If you want the answer let me know...I'm not keeping it from you, but if you want to work it out, lets start with this:

    For k = 1 the graph should look like this:

    .....♥
    .♥ ♦ ♥
    .....♥

    With 1 step

    The ♥ are the spots you can get to, the ♦ are spots you can't get to.

    N_s = 4


    ( You don't have to make a picture, but if you wish to this is where I got the ...[text shortened]... - https://www.alt-codes.net/ just click them and they copy, then just paste them )

    For k= 2 ?
    I think it's time for the answer on this one please.
  5. R
    Standard memberRemoved
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    09 Mar '21 19:472 edits
    @venda said
    I think it's time for the answer on this one please.
    General Solution for the number of places you can reach in exactly "k" steps

    N_k = ( k+1)²

    k = 2
    N_2 = ( 2+1 )² = 9

    .......♥
    ....♥♦♥
    .♥♦♥♦♥
    ....♥♦♥
    .......♥

    k = 3
    N_3 = ( 3+1 )² = 16

    ..........♥
    .......♥♦♥
    ....♥♦♥♦♥
    .♥♦♥♦♥♦♥
    ....♥♦♥♦♥
    .......♥♦♥
    ..........♥


    The general proof is a more involved, but I think you can verify and notice the pattern. continue if you'd like. If your happy, I'm happy. However, if you want to see the general proof for all "k" let me know. Its just going to be ugly because of inability to format math notation properly. I wish this forum supported LateX.

    🙂
  6. Subscribervenda
    Dave
    S.Yorks.England
    Joined
    18 Apr '10
    Moves
    83457
    09 Mar '21 20:29
    @joe-shmo said
    General Solution for the number of places you can reach in exactly "k" steps

    N_k = ( k+1)²

    k = 2
    N_2 = ( 2+1 )² = 9

    .......♥
    ....♥♦♥
    .♥♦♥♦♥
    ....♥♦♥
    .......♥

    k = 3
    N_3 = ( 3+1 )² = 16

    ..........♥
    .......♥♦♥
    ....♥♦♥♦♥
    .♥♦♥♦♥♦♥
    ....♥♦♥♦♥
    .......♥♦♥
    ..........♥


    The general proof is a more involved, but I thik you can verift and notice the patte ...[text shortened]... y because of inability to format math notation properly. I wish this forum supported LateX.

    🙂
    No, that'll do thanks.
    I appreciate all the time you spend trying to educate people such as myself!!
    On the positive side,judging by the responses we see ,most people don't even have a go!
  7. R
    Standard memberRemoved
    Joined
    10 Dec '06
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    8528
    09 Mar '21 20:321 edit
    @venda said
    No, that'll do thanks.
    I appreciate all the time you spend trying to educate people such as myself!!
    On the positive side,judging by the responses we see ,most people don't even have a go!
    We all here learning...what you get out is proportional to what you put in ( in some way )
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