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23 members have voted

gordonm888
Joined: Feb 18, 2015
• Posts: 2848
February 23rd, 2021 at 1:19:10 PM permalink
Several years ago I started analyzing the frequency with which words appeared in Boggle, and ever since I have been playing around with the combination math involved in placing strings of characters into grids of finite size, i.e. grids with edges. The puzzle below, which is one of the simplest problems of this kind, will serve to illustrate what this means.

Given a 3x3 grid of 9 squares; and each square is considered to be connected to other squares in the grid by their face or by their corners, as in the game Boggle. Thus, the center square is connected to the other 8 squares, corner squares are connected to 3 squares and edge squares are connected to 5 squares.

Now consider a string of 3 characters. Let's make the characters letters and the string a 3-letter word, to continue to emulate the game Boggle. When placing a 3-letter word into the 3x3 grid the rules are:

1. The other six squares will be blank
2. The string of letters must be arranged in the grid on connected squares such that they can be read in the correct sequence to form the word.
3. No square in the grid can be used twice in “spelling” the word.
4. For a word that is a palindrome such as “WOW” the fact that it can be read forwards and backwards does not mean that it can be counted twice; any distinct arrangement of a palindromic word counts as one single arrangement.
Note: Rules 2-4 are standard Boggle rules.

How many distinct arrangements of the following three letter words can be formed in a 3x3 grid of 9 squares?
SEX
EEL
OFF
WOW
ZZZ
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
gordonm888
Joined: Feb 18, 2015
• Posts: 2848
February 23rd, 2021 at 6:43:34 PM permalink
I think I killed the thread.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
chevy
Joined: Apr 15, 2011
• Posts: 84
February 23rd, 2021 at 6:51:50 PM permalink
Quote: gordonm888

How many distinct arrangements of the following three letter words can be formed in a 3x3 grid of 9 squares?
SEX
EEL
OFF
WOW
ZZZ

I admit to not understanding what you are asking. I count at least 13 letters needed (SEXELOFFWWZZZ) to form the 5 words, but only a 3x3=9 grid.

Obviously I am missing something (After I get past that hurdle, then I can fall back on just not being able to solve)
chevy
Joined: Apr 15, 2011
• Posts: 84
February 23rd, 2021 at 7:23:49 PM permalink

Unless??????????
Are we supposed to fit one word into the 3x3 grid in as many ways as possible

For example SEX

S corner : 15 ways * 4 corners = 60
S middle of side : 19 ways * 4 middles = 76
S center : 16 ways

Total = 152

Or if orientation of the board does not matter
SEX on top

same as

X
E
S on side

Then 15 + 19 + 4 = 38

?????????????

Then repeat for the other words?
ThatDonGuy
Joined: Jun 22, 2011
• Posts: 4667
Thanks for this post from:
February 23rd, 2021 at 7:26:19 PM permalink
Quote: chevy

Quote: gordonm888

How many distinct arrangements of the following three letter words can be formed in a 3x3 grid of 9 squares?
SEX
EEL
OFF
WOW
ZZZ

I admit to not understanding what you are asking. I count at least 13 letters needed (SEXELOFFWWZZZ) to form the 5 words, but only a 3x3=9 grid.

Obviously I am missing something (After I get past that hurdle, then I can fall back on just not being able to solve)

What I think he means is this:
How many ways can each of the three sets of letters be placed in a 3x3 grid so the second letter is adjacent to the first and the third is adjacent to the second? Each one is meant to be treated separately.

I assume that EEL and OFF have the same number, since each spelling of EEL can be changed to one of OFF by replacing the L with an O, the Es with Fs, and going in the reverse direction.
chevy
Joined: Apr 15, 2011
• Posts: 84
February 23rd, 2021 at 7:41:32 PM permalink
Quote: ThatDonGuy

What I think he means is this:
How many ways can each of the three sets of letters be placed in a 3x3 grid so the second letter is adjacent to the first and the third is adjacent to the second? Each one is meant to be treated separately.

I assume that EEL and OFF have the same number, since each spelling of EEL can be changed to one of OFF by replacing the L with an O, the Es with Fs, and going in the reverse direction.

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

chevy
Joined: Apr 15, 2011
• Posts: 84
February 23rd, 2021 at 8:02:19 PM permalink

If orientation and sequence matter I get the same for EEL and SEX, so I will assume they don't as it makes the problem more interesting. Listing ways with first letter in corner, side middle, and center...not counting duplicate triples.

SEX = 15 + 19 + 4 = 38 ways???
EEL = 15 + 13 + 0 = 28 ways???

OFF = 28 per ThatDonGuy's comment.

ZZZ= 10 + 2 + 0 = 12 ways ???

charliepatrick
Joined: Jun 17, 2011
• Posts: 2335
February 24th, 2021 at 8:18:59 AM permalink
Quote: gordonm888

...
How many distinct arrangements of the following three letter words can be formed in a 3x3 grid of 9 squares?
SEX
EEL
OFF
WOW
ZZZ

For simplicity assume the squares are:-
`A B CD E FG 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

chevy
Joined: Apr 15, 2011
• Posts: 84
February 24th, 2021 at 9:55:04 AM permalink

Your solution is more elegant than me. I did the problem by a more brute force approach. I think I get similar answers for EEL/OFF and ZZZ.

But if we consider SEX....I considered possibilities with S in corner, S in Middle of a side, S in center....I can't find where I come up short of your number.

using you grid labels

corner: (15 ways)
S at A, E can be B.....4 choices for X....(D,E,F,C)
S at A, E can be D.....4 choices for X....(B,E,H,G)
S at A, E can be E.....7 choices for X....(B,C,F,J,H,G,D)

middle of side : (19 ways)
S at B, E can be at A......2 choices for X.....(D,E)
S at B, E can be at C......2 choices for X.....(E,F)
S at B, E can be at D......4 choices for X.....(A,E,H,G)
S at B, E can be at F......4 choices for X.....(C,E,H,J)
S at B, E can be at E.....7 choices for X.....(A,D,G,H,J,F,C)

center : (16 ways)
S at E, E can be at the other 8 locations, and X can be either side of E (around the perimeter)

Total ways to have SEX
=4 corners + 4 middles sides + 1 center
=4*15+4*19+16
=152 ways

Anybody see where I am missing some SEX.....8 of them apparently???

charliepatrick
Joined: Jun 17, 2011