#C6.txt: Maple code for Lecture 6 of ExpMath by Dr.Z.

Help6:=proc(): print(`LD(L) , Walks2D(A,pt) , NuWalks2D(A,pt), FootBall()`): 
print(`RandWalk2D(A,pt)`):
end:

#Walks2D(A,pt): Inputs a list A of `atomic' steps A
#in the form [a,b] (with a,b non-neg. and a+b>0) and
#a final location (final score in game, e.g. [31,9]).
#OUTPUTS the SET of all WALKS from [0,0] to
#to pt, described by giving the intermedite location
#[[0,0],[2,0],[10,0],[10,2],[10,10]]
Walks2D:=proc(A,pt) local i,W,W1,w:
option remember:

if pt[1]<0 or pt[2]<0 then
 RETURN({}):
fi:

if pt=[0,0] then
 RETURN({[[0,0]]}):
fi:

W:={}:

for i from 1 to nops(A) do
 W1:=Walks2D(A,pt-A[i]):
 W:=W union {seq([op(w),pt], w in W1)}:
od:

W:
end:

FootBall:=proc():
[[0,2],[0,3],[0,6],[0,7],[0,8],[2,0],[3,0],[6,0],[7,0],[8,0]]:
end:

#NuWalks2D(A,pt): Inputs a list A of `atomic' steps A
#in the form [a,b] (with a,b non-neg. and a+b>0) and
#a final location (final score in game, e.g. [31,9]).
#OUTPUTS the NUMBER of all WALKS from [0,0] to
#to pt, described by giving the intermedite location
#[[0,0],[2,0],[10,0],[10,2],[10,10]]
NuWalks2D:=proc(A,pt) local i,W,W1,w:
option remember:

if pt[1]<0 or pt[2]<0 then
 RETURN(0):
fi:

if pt=[0,0] then
 RETURN(1):
fi:

W:=0:

for i from 1 to nops(A) do
 W1:=NuWalks2D(A,pt-A[i]):
 W:=W + W1:
od:

W:
end:

#RandWalk2D(A,pt): inputs a list of atomic steps A and 
# a final destination pt, outputs UNIFORMLY at random
#a walk from [0,0] to pt using the steps in A
RandWalk2D:=proc(A,pt) local i,W1,Die1,r:
if pt[1]<0 or pt[2]<0 then
 RETURN(FAIL):
fi:

if pt=[0,0] then
   RETURN([[0,0]]):
fi:

Die1:=[seq(NuWalks2D(A,pt-A[i]),i=1..nops(A))]:

#Let r be the index in A of the last step in our uniform-random walk
#that ended at pt 
r:=LD(Die1):

W1:=RandWalk2D(A,pt-A[r]):

[op(W1),pt]:

end:

 ##start GOOD (subdiagonal walks)
#GWalks2D(A,pt): Inputs a list A of `atomic' steps A
#in the form [a,b] (with a,b non-neg. and a+b>0) and
#a final location (final score in game, e.g. [31,9]).
#OUTPUTS the SET of all GOOD (subdiagonal) WALKS from [0,0] to
#to pt, described by giving the intermedite location
#[[0,0],[2,0],[10,0],[10,2],[10,10]]
GWalks2D:=proc(A,pt) local i,W,W1,w:
option remember:

if (pt[1]<0 or pt[2]<0 or pt[1]<pt[2]) then
 RETURN({}):
fi:

if pt=[0,0] then
 RETURN({[[0,0]]}):
fi:

W:={}:

for i from 1 to nops(A) do
 W1:=GWalks2D(A,pt-A[i]):
 W:=W union {seq([op(w),pt], w in W1)}:
od:

W:
end:

FootBall:=proc():
[[0,2],[0,3],[0,6],[0,7],[0,8],[2,0],[3,0],[6,0],[7,0],[8,0]]:
end:




##FROM C4.txt
#LD(L): Inputs a list of positive integers L (of n:=nops(L) members)
#outputs an integer i from 1 to n with the prob. of i being
#proportional to L[i]
#For example LD([1,2,3]) should output 1 with prob. 1/6
#output 2 with prob. 1/3
#output 3 with prob. 3/6=1/2
LD:=proc(L) local n,i,su,r:
n:=nops(L):
r:=rand(1..convert(L,`+`))():
su:=0:

for i from 1 to n do
 su:=su+L[i]:
  if r<=su then
    RETURN(i):
  fi:
od:
end:
#END FROM C4.txt