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 Class Pass (Posted on 2004-06-24)
The teacher in a certain class room allows you to pass a paper with an assignment around, and whomever it ends up on has to do it. The only two rules are you can't pass it to someone who already has had it and you can only pass it to the person to the left, right, forward, or backward.

In a room of 30 students arranged in a 6 by 5 grid, the teacher starts out with the assignment somewhere on the front row of 6 students. At some point someone is stuck holding the assignment because all his neighbors have had it and passed it on to someone else. If this happens after every student in the room has had it, what is the probablity, for each individual, that he or she turns out to be the lucky winner of the assignment?

 No Solution Yet Submitted by Gamer Rating: 4.0000 (2 votes)

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 Better randomization for simulation. | Comment 6 of 15 |

This version of the simulation has a different algorithm for randomizing the direction, changed on the theory that the second call to RND might be influence by the first or some such thing.  The results look more symmetric now:

DEFINT A-Z
DIM drCt(-2 TO 2) AS LONG
DIM dcCt(-2 TO 2) AS LONG
RANDOMIZE TIMER
DO
REDIM student(6, 7)' unused elements outside bound
f = INT(RND(1) * 6 + 1)
student(1, f) = 1
row = 1: col = f
FOR pass = 2 TO 30
DO
DO
rNo = INT(RND(1) * 9)
dr = rNo \ 3 - 1
dc = rNo MOD 3 - 1
LOOP UNTIL (dr OR dc) AND (dr = 0 OR dc = 0)
drCt(dr) = drCt(dr) + 1
dcCt(dc) = dcCt(dc) + 1
newRow = row + dr: newCol = col + dc
LOOP UNTIL newRow > 0 AND newRow < 6 AND newCol > 0 AND newCol < 7 AND student(newRow, newCol) = 0
row = newRow: col = newCol
p = pass
surr = 1
FOR dr = -1 TO 1
FOR dc = -1 TO 1
IF ABS(dr) <> ABS(dc) THEN
newRow = row + dr: newCol = col + dc
IF newRow > 0 AND newRow < 6 AND newCol > 0 AND newCol < 7 THEN
IF student(newRow, newCol) = 0 THEN
surr = 0: EXIT FOR
END IF
END IF
END IF
NEXT
IF surr = 0 THEN EXIT FOR
NEXT
IF surr THEN EXIT FOR
student(row, col) = 1
NEXT

IF p = 30 THEN
tot(row, col) = tot(row, col) + 1: overTot = overTot + 1
fCount = fCount + 1
PRINT row, col, fCount, tr&
' FOR i = -2 TO 2: PRINT drCt(i); : NEXT
' PRINT
' FOR i = -2 TO 2: PRINT dcCt(i); : NEXT
' PRINT
END IF
tr& = tr& + 1
LOOP UNTIL fCount = 10000
FOR row = 1 TO 5
FOR col = 1 TO 6
PRINT USING "#####"; tot(row, col);
NEXT
PRINT
NEXT
PRINT
FOR row = 1 TO 5
FOR col = 1 TO 3
PRINT USING "#####"; tot(row, col) + tot(row, 7 - col);
NEXT
PRINT
NEXT
PRINT
PRINT tr&

In a sample run it took 1,660,656 trials to get 10,000 in which all 30 students held the assignment.

The array of final hits per seat is:

` 673  302  409  384  261  813 187  550  318  226  308  205 275  225  278  271  261  468 112  440  303  285  548  121 494  111  195  275   97  605`

Which is much more nearly symmetric.

Combining the left and right corresponding number, and plotting on the left, the totals are:

`1486  563  793 392  858  544 743  486  549 233  988  5881099  208  470`

Implying the percentages, rounded to the nearest percent are half of the below figures (as it's based on the number combined left and right, and duplicated):

`15  6  8  8  6 15 4  9  5  5  9  4 7  5  5  5  5  7 2 10  6  6 10  211  2  5  5  2 11`

So the most dangerous places are the corners of the room, with about 7 1/2% probability for each of the front-left and front-right seats, and 5 1/2% each for rear-left and rear-right.  The safest places are just to the front of or next to the guy in a back corner--only about 1% chance of landing there.

Of course this counts only those cases which extend all the way to having everyone having seen the paper, and so is a conditional probability.  In only about 1 in 166 trials does this happen altogether.

If we count all the cases when the passing is forced to stop by the holder being surrounded by previous seers, we get:

` 919  400  420  410  396  969 210  359  252  207  341  205 243  248  194  228  221  223 181  309  204  206  341  166 780  116  167  198  129  758`
`1888  796  830 415  700  459 466  469  422 347  650  4101538  245  365`

For percentages that are half of:

`19  8  8  8  8 19 4  7  5  5  7  4 5  5  4  4  5  5 3  7  4  4  7  315  2  4  4  2 15`
` `

 Posted by Charlie on 2004-06-24 21:56:34

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