3 edits
@venda saidIf 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:
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!!
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 ?
@joe-shmo saidI can see where this is going I think.
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'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
@joe-shmo saidI think it's time for the answer on this one please.
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 ?
2 edits
@venda saidGeneral Solution for the number of places you can reach in exactly "k" steps
I think it's time for the answer on this one please.
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.
🙂
@joe-shmo saidNo, that'll do thanks.
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.
🙂
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!
1 edit
@venda saidWe all here learning...what you get out is proportional to what you put in ( in some way )
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!