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gordonm888
gordonm888
Joined: Feb 18, 2015
  • Threads: 44
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February 24th, 2021 at 3:06:58 PM permalink
Quote: chevy

Okay, that makes sense. Two more questions


1) Does orientation matter?
SEX. (top)
vs
X
E
S
on left side
2) Palindromes ruled out....but what about same 3 squares in different order?
Squares 1,2,4

E E
L

Form EEL by 124 and 214



Orientation does not matter. Words can be read from the bottom up or backwards or forwards, etc. They may be twisted into what I call a triangle, such as

SX
E

or

XE
S

or whatever.

Palindromes simply lack the ability to be sequenced backwards in a particular set of squares because that would not count as a distinct arrangement with respect to being sequenced forwards.

You are correct about off and eel having the same number of arrangements. I listed both of them just to have the complete set of symmetries for a 3-character string.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
gordonm888
gordonm888
Joined: Feb 18, 2015
  • Threads: 44
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February 24th, 2021 at 3:07:01 PM permalink
Quote: chevy

Okay, that makes sense. Two more questions


1) Does orientation matter?
SEX. (top)
vs
X
E
S
on left side
2) Palindromes ruled out....but what about same 3 squares in different order?
Squares 1,2,4

E E
L

Form EEL by 124 and 214



Orientation does not matter. Words can be read from the bottom up or backwards or forwards, etc. They may be twisted into what I call a triangle, such as

SX
E

or

XE
S

or whatever.

Palindromes simply lack the ability to be sequenced backwards in a particular set of squares because that would not count as a distinct arrangement with respect to being sequenced forwards.

You are correct about off and eel having the same number of arrangements. I listed both of them just to have the complete set of symmetries for a 3-character string.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
gordonm888
gordonm888
Joined: Feb 18, 2015
  • Threads: 44
  • Posts: 2809
February 24th, 2021 at 3:16:42 PM permalink
Quote: charliepatrick

For simplicity assume the squares are:-
A B C
D E F
G H J
Then there are 84 sets of three squares. They fall into three types
(i) Linear - e.g. A-B-C - you can only go in a forwards or backwards direction as you cannot reach A from C - so ABC or CBA.
(ii) Circular - e.g. A-B-D - you can reach A from D so they form a circle (or triangle) - so ABD BDA DAB DBA BAD ADB.
(iii) Impossible - e.g. A-B G - you cannot form a joined triangle. You only need these to check the total combinations is correct. (I couldn't see a mathematical way of working out the combinations, so just listed them all. With larger numbers one might write a program to go through them, but then there would be more "shapes" than linear and circular.)

Linear
SEX/EEL/OFF - you can go forwards or backwards (2)
WOW/ZZZ - you can only go forwards (as the backwards is an identical placement of letters) (1)
Circular
SEX - where all the letters are different you can use any of the six ways (6)
EEL/OFF/WOW - where one of the letters is different it can be in any of three places (3)
ZZZ - where all the letters are different only one comination counts (1)
NumberTriple typeSEXEELOFFWOWZZZ
32
Linear
2
2
2
1
1
16
Circular
6
3
3
3
1
160
112
112
80
48



This is absolutely CORRECT, and using the same methodology that I use.


The methodology used by Charlie allows you to straightforwardly address how many three character strings would be on a 4x4 grid or more generally, on an n x n grid.

This methodology is more complex when you are analyzing strings that are 4, 5 and 6 characters in length because there are far more spatial arrangements that the string can be twisted into, especially for 4x4 and larger grids, Also there are far more symmetrical arrangements for repeated characters in the strings that need to be considered. I have never attempted a 7 character string.

And I really wish I was smart enough to figure out an approach for calculating how many ways a 9 character string can be arranged in a 3x3 grid. I've never really seriously attempted that.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
chevy
chevy
Joined: Apr 15, 2011
  • Threads: 2
  • Posts: 75
February 24th, 2021 at 3:32:49 PM permalink
Quote: charliepatrick

If you go from E to A, then the next place can only be D or B. However if you go from E to B, then you can go to A C D or F. This adds two more for EB, and similarly for ED EF and EH.



Thank you very much!!! I guess I ended my counting of SEX from the center square prematurely.
charliepatrick
charliepatrick
Joined: Jun 17, 2011
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February 25th, 2021 at 2:11:12 AM permalink
Quote: gordonm888

....I really wish I was smart enough to figure out an approach for calculating how many ways a 9 character string can be arranged in a 3x3 grid....

I didn't know how to do this so initially did it by hand. However when the alternative answer of 152 came in I couldn't see the 8 missing ones, so only then knocked up a program to see which answer was correct. This could easily be extended for larger grids but doesn't find nor define which ones have which patterns. It can only really be used to confirm the total number of valid strings.
(Some lines omitted - for larger grids I'd do it the other way round as there are fewer valid links than invalid ones)

n=0;
for (i=1; i<=9; i++)
{
for (j=1; j<=9; j++)
{
for (k=1; k<=9; k++)
{
invalid=0;
if (i==j) {invalid=1;};
if (i==k) {invalid=1;};
if (j==k) {invalid=1;};
/* Check first move is legal */
if ((i==1)&&(j==3)) {invalid=1;};
if ((i==1)&&(j==6)) {invalid=1;};
etc.
if ((i==9)&&(j==4)) {invalid=1;};
if ((i==9)&&(j==7)) {invalid=1;};

/* Check second move is legal */
if ((j==1)&&(k==3)) {invalid=1;};
if ((j==1)&&(k==6)) {invalid=1;};
etc.
if ((j==9)&&(k==4)) {invalid=1;};
if ((j==9)&&(k==7)) {invalid=1;};

if (invalid==0)
{
n++;
outline="N("+n+") ";
outline+="I: "+i+" ";
outline+="J: "+j+" ";
outline+="K: "+k+" ";
outline+="<BR>";
document.write(outline);
};
};
};
};
Gialmere
Gialmere
Joined: Nov 26, 2018
  • Threads: 39
  • Posts: 1821
February 25th, 2021 at 8:28:54 AM permalink


You don't play golf but decide to try a bizarre online system for the game that uses meditation and self hypnosis.

Before playing, you intensely concentrate on two strokes of different lengths to use for the entire round, one a drive, the other an approach, and play directly toward the hole so that various combinations of the two distances will get you there.

The ball must go the full length on each stroke, but you may go beyond the hole with either stroke, then play back toward the hole. All strokes are on a straight line toward the hole and will go in the cup if the stroke and hole distances are equal.

Using this odd technique of selecting two exact distances only, what is your lowest score possible on the following nine-hole course?


Hole Length Score
#1 150 yds ?
#2 300 yds ?
#3 250 yds ?
#4 325 yds ?
#5 275 yds ?
#6 350 yds ?
#7 225 yds ?
#8 450 yds ?
#9 425 yds ?
Have you tried 22 tonight? I said 22.
Joeman
Joeman
Joined: Feb 21, 2014
  • Threads: 35
  • Posts: 1897
February 25th, 2021 at 9:20:39 AM permalink
I can get down to 28 strokes.

Similar to my actual golf game, my Drive would be 150 yd, and my Approach would be 25 yd. Out in 28:

Hole Length Score Shots
#1 150 yds 1 150
#2 300 yds 2 150 + 150
#3 250 yds 4 150 + 150 - 25 - 25
#4 325 yds 3 150 + 150 + 25
#5 275 yds 3 150 + 150 - 25
#6 350 yds 4 150 + 150 + 25 + 25
#7 225 yds 4 150 + 25 + 25 + 25
#8 450 yds 3 150 + 150 + 150
#9 425 yds 4 150 + 150 + 150 - 25
"Dealer has 'rock'... Pay 'paper!'"
charliepatrick
charliepatrick
Joined: Jun 17, 2011
  • Threads: 33
  • Posts: 2322
Thanks for this post from:
Gialmere
February 25th, 2021 at 9:23:34 AM permalink
I'll have to think about an analytical approach to this but I did like my first guess....
Dividing all the lengths by 25 gets 6 12 10 13 11 14 9 17 18 = 6 17 18 (9 thru 14). 7 and 3 (i.e. 175 and 75 yards) gives 26 shots. (2 4 2 3 3 2 3 3 4).
Wizard
Administrator
Wizard
Joined: Oct 14, 2009
  • Threads: 1367
  • Posts: 22723
February 25th, 2021 at 9:28:53 AM permalink
Chag Purim Sameach!, my fellow math enthusiasts.

Here is my guess,

30



75 and 125 yards. I can complete the course with a score of 30.

I am not saying this is optimal, but have a good feeling about it.

I first divided all the yardages by 25 to get:

6
12
10
13
11
14
9
18
17

Then I just know somewhere that primes are good for constructing other numbers. With a little trial and error 3 and 5 seem to work well.

For example,
hole 1: 3+3
hole 2: 5+5+5-3
hole 3: 5+5

Can anyone beat my 30?

It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
Administrator
Wizard
Joined: Oct 14, 2009
  • Threads: 1367
  • Posts: 22723
February 25th, 2021 at 9:55:42 AM permalink
Looks like Charlie is winning the WoV golf match so far and I'm losing, which is usually the case when I golf.
It's not whether you win or lose; it's whether or not you had a good bet.

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