GingerAle

ok, I'm not a mind reader.  

Actually, you are ...in a metaphorical sense. You’ve demonstrated such skills on more than one occasion. I think most teachers need above average skills in mind reading to be a good teacher. Beyond basic mind-reading is the quintessential skill of filling in the thoughts for thought processes that are missing essential information in the student’s mind.

For teaching younger students, this is a common and relatively easy requirement because younger students are usually just missing information. The difficulty is in persuading the student to use his/her mind to its full capacity.

For the older student, however, the requirements are more arduous because wrong and errant information may be firmly entrenched. The student needs to be willing to excise errant thought process to make room for the correct processes.     

There is a valid argument that to read a mind, there needs to be a mind to read. Mindlessness is a real phenomenon. It’s exceptionally difficult to comprehend and hence exceptionally difficult to treat. 

GA

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Solution: \(\dfrac{5!}{2!} * \dbinom {6}{3}\)

Description of process:

Removing the ‘C’s from the set gives (5!/2! =60) distinguishable arraignments for the remaining five (5) letters. The three ‘C’s can then occupy three (3) of the six (6) positions between the letters and at either end of the distinguishable arraignments

Graphic demonstrating positions:  _A_U_R_A_Y_   

Select the three (3) positions as \(\dbinom {6}{3} = 20\)

Total number of arrangements where no ‘C’s are adjacent to another ‘C’ \(= (60 * 20) = 1200 \)

GA

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Hi Melody,

OMG! I saw two trees but missed the whole bloody forest.

[You’re in a flood, but I’m the one that’s all wet...]

Until I saw your graph, I had tunnel vision. I didn’t really do a (full) positional analysis. I only compared the positions of the two sets for each simultaneous sequence.   The two diagonals are the only two that occur simultaneously in the 8! counts.

It’s Spooky Quantum Dumbness at a distance and I walked right into it, face first, because of quantum blindness.

Lancelot Link would throw banana peels and coconuts at me for that.  

Very Cool! Thank you.

GA

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Hi Melody,

 But I have to subtract all the ones that are in both every row and every column (becasue they have been added twice.)This number is 8!

You’ve subtracted (8!) from the total successful arrangements. That is a lot duplication that is NOT indicated in the positional analysis of the rooks. The only time there is a rook in every row and every column is when they are positioned diagonally across the board.  Other than this, one or more rows and/or one more columns are unoccupied, but not unattacked.   

As I described above, this diagonal arraignment occurs twice: once, descending from the top right to the bottom left, and again from the bottom left to the top right. These diagonal arraignments are the ONLY two arrangements that occur in both the vertical movement progression and the horizontal movement progression.  There are no other duplicates, so where does the (8!) come from?

Appended to this post are images depicting the chessboard with eight rooks in horizontal and vertical formats, demonstrating their respective movements on the chessboard and points of interest.

If I am missing (8!-2) arrangements where these two formats intersect, then I have royally rooked myself. I would really like to see them. Can you graphically demonstrate where some of these duplications occur?

GA

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I agree with your interpretation, Ron:

It’s a quadratic expression divided by a linear expression.

\(\large \frac{8x^2 - 4x} {4x – 2} \rightarrow 2x\)     This is implied by the exception "that x isnt equal to 1/2" (sic).

However, because it is written in ASCII SLOP it needs parenthetical operators to confirm this, else it can (and should) be strictly interpreted as:

\(\large 8x^2 – 4*\frac{x}{4}* x – 2 \rightarrow 7x^2 -2 \)

Both the forum’s calculator and Wolfram Alpha interpret the expression using Formal ASCII Mathematical Hierarchy.  There are good reasons why ASCII Mathematical Hierarchy was formalized by mathematicians, engineers, and computer scientists. ASCII was developed to communicate with computers, not to teach math to humans; Eventually its use evolved into communicating with humans ...and genetically enhanced Chimps...and along with it came ambiguity because the composers or the readers do not know the Mathematical Hierarchy.

Of course computers can be programmed to make assumptions –ignoring hierarchy; so there are online algebra solvers that will simplify or solve it as a quadratic expression divided by a linear expression, without the use of parenthetical operators. 

GA

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Hello CPhill,

When calculating a statistic on indistinguishable elements or arraignments, they are (usually) treated as if they are distinguishable.  Though not distinguishable as one from another, their appearance is proportional to their multiplicity, and this affects their appearance counts in statistical measures.  

GA

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There is no casual solution to this problem; though the solution process is not ultra complex.

The use of the Inclusion-Exclusion principle will accelerate the process. Like usual, its application requires careful and (sometimes) laborious attention to avoid over-counting successes and failures.

The question resolves to a binary state of either an arrangement of eight (8) rooks occupying or attacking every square, or it does not. Analyzing the counts of unoccupied and un-attacked squares is not requires for this question.

(Permutations of the rooks among themselves are not necessary, as this will be done for both success and failure counts and will factor out.) 

Setup:

Vertical format.

 Starting with an 8 X 8 chessboard place eight rooks on the top row (row 1) and observe that every square is either occupied or attacked by a rook. Moving the rooks vertically (row-wise) still allows for every square to be either occupied or attacked by a rook. There are (8^8) unique positions when moving the rooks vertically.

This same sequence is repeatable in a horizontal format.

Place the eight rooks in the left column (Column 1) observe that every square is either occupied or attacked by a rook. Moving the rooks horizontally (column-wise) still allows for every square to be either occupied or attacked by a rook. There are (8^8) unique positions when moving the rooks vertically. HOWEVER, in this horizontal format, there are two (2) cases where the positions of the rooks identically match the positions in the vertical format. These two (2) cases occur when the rooks align diagonal positions across the board.  This occurs twice: from the top-left to the bottom-right, and from the bottom-left to the top-right.  These two (2) cases are exclusions and need to be subtracted from the total. So at this point, the subtotal for success counts are (8^8) + [(8^8) – (2)].

At this point it’s (apparently) discernable that for every square to be either occupied or attacked by a rook, every column or every row must have at least one rook. Removing all rooks from a column leaves some (not all) row squares on that column un-attacked.  Like-wise, removing all rooks from a row leaves some (not all) column squares on that row un-attacked.   

If the above is true, then these counts (8^8) + [(8^8) – (2)] are the only successes.

Probability of a random arrangement of eight (8) rooks to occupy or attack every square on a chessboard:

\(\rho_{(s)} = \dfrac {(2*8^8)-2} {\binom {64}{8}} \approx 0.75809\%\)

GA

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Is this really your calculation? It looks like you copied it from somewhere.

If it is yours, then how did you formulate your calculation?

It is unlikely what you presented is the correct probablity.

Re-Present your calculation as an equation with the work product: delineate your calculations showing how the Inclusion-Exclusion principle applies to your calculation. Specifically, define the included set, the excluded set, and the union of these two sets. 

If you do this, then it should be easy for someone to verify your solution, without deriving and solving this. (As an analogy, it’s easy to verify the primes of large numbers; it’s not so easy to find the primes in the first place.) 

If you do decide to do this, use LaTex. Reading math in ASCII slop is often irritating.

GA

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