*Introduction to Evolutionary Informatics,*by Robert J. Marks II, the “Charles Darwin of Intelligent Design”; William A. Dembski, the “Isaac Newton of Information Theory”; and Winston Ewert, the “Charles Ingram of Active Information.” World Scientific, 332 pages.

Classification: Engineering mathematics. Engineering analysis. (TA347)

Subjects: Evolutionary computation. Information technology–Mathematics.

Subjects: Evolutionary computation. Information technology–Mathematics.

^{1}*Cracker Barrel puzzle*in section 5.4.1.2

*Endogenous information of the Cracker Barrel puzzle*(p. 128). The rules of this variant of a triangular peg-solitaire are described in the text (or can be found at wikipedia's article on the subject). The humble authors then describe a simulation of the game to calculate how probable it is to solve the puzzle using moves at random:

A search typically requires initialization. For the Cracker Barrel puzzle, all of the 15 holes are filled with pegs and, at random, a single peg is removed. This starts the game. Using random initialization and random moves, simulation of four million games using a computer program resulted in an estimated win probability p = 0.0070 and an endogenous information of $$I_\Omega = -\log_2 p = 7.15 bits.$$ Winning the puzzle using random moves with a randomly chosen initialization (the choice of the empty hole at the start of the game) is thus a bit more difficult than flipping a coin seven times and getting seven heads in a rowNaturally, I created such an simulation in R for myself: I encoded all thirty-six moves that could occur in a matrix

`cb.moves`

, each row indicating the jumping peck, the peck which is jumped over, and the place on which the peck lands.
And here is the little function which simulates a single random game:
```
cb.simul <- function(pos){
```

# pos: boolean vector of length 15 indating position of pecks

# a move is allowed if there is a peck at the start position & on the field which is

# jumped over, but not at the final position

allowed.moves <- pos[cb.moves[,1]] & pos[cb.moves[,2]] & (!pos[cb.moves[,3]])

# if now move is allowed, return number of pecks left

if(sum(allowed.moves)==0) return(sum(pos))

# otherwise, chose an allowed move at random

number.of.move <- ((1:36)[allowed.moves])[sample(1:sum(allowed.moves),1)]

pos[cb.moves[number.of.move,]] <- c(FALSE,FALSE,TRUE)

return(cb.simul(pos))

}

I run the simulation 4,000,000 times, changing the start position at random. But as a result, my estimated win probability was $p_e=0.0045$ - only two thirds of the number in the text. How can this be? Why were Prof. Marks et.al. so much luckier than I? I re-run the simulation, checked the code, washed, rinsed, repeated: no fundamental change.
So, I decided to take a look at all possible games and on the probability with which they occur. The result was this little routine:```
cb.eval <-
function(pos, prob){
```

#pos: boolean vector of length 15 indicating position of pecks

#prob: the probability with which this state occurs
# a move is allowed if there is a peck at the start position & on the field which is

#jumped over, but not at the final position

allowed.moves <- pos[cb.moves[,1]] & pos[cb.moves[,2]] & (!pos[cb.moves[,3]])

if(sum(allowed.moves)==0){

#end of a game: prob now holds the probability that this game is played
nr.of.pecks <- sum(pos)

#number of remaining pecks
cb.number[nr.of.pecks] <<- cb.number[nr.of.pecks]+1

#the number of remaining pecks is stored in a global variable
cb.prob[nr.of.pecks] <<- cb.prob[nr.of.pecks] + prob

#the probability of this game is added to the appropriate place of the global variable

return()

}

for(k in 1:sum(allowed.moves)){

#moves are still possible, for each move the next stage will be calculated
d <- pos

number.of.move <- ((1:36)[allowed.moves])[k]

d[cb.moves[number.of.move,]] <- c(FALSE,FALSE,TRUE)

cb.eval(d,prob/sum(allowed.moves))

}

}

I now calculated the probabilities for solving the puzzle for each of the fifteen possible starting positions. The result was $$p_s=0.0045 .$$This fits my simulation, but not the one of our esteemed and humble authors! What had happened?
## An educated guess

I found it odd that the authors run 4,000,000 simulations - 1.000,000 or 10,000,000 seem to be more commonly used numbers. But when you look at the puzzle, you see that it was not necessary for me to look at all fifteen possible starting positions - whether the first peck is missing in position 1 or position 11 does not change the quality of the game: you could rotate the board and perform the same moves. Using symmetries, you find that there are only four essentially different starting positions. the black, red, and blue group with three positions each, and the green group with six positions. For each group, you get a different probability of successgroup | black | green | red | blue |

prob. of choosing this group | .2 | .4 | .2 | .2 |

prob. of success | .00686 | .00343 | .00709 | .001726 |