with(group):
G:=permgroup(36,{a=[[1,2]], b=[[seq(i,i=1..36)]]}):

moves:=proc(zz)
local perm,cycle,j,nummoves,nextcard,foundmate;
#
perm:=array[1..36]:
for cycle in zz do 
	for j from 1 to nops(cycle)-1 do perm[cycle[j]]:=cycle[j+1]:od:
	perm[cycle[nops(cycle)]]:=cycle[1]:
od:
#
nummoves:=0:
nextcard:=1:
#
while nextcard<=36 do
	foundmate:=FALSE:
	for j from 1 to nextcard-1 do 
		if perm[nextcard]=perm[j] mod 18 then foundmate:=TRUE:fi:
	od:
	if foundmate=FALSE then 
		nextcard:=nextcard+1:
		foundmate:=FALSE:
		for j from 1 to nextcard-1 do 
		   if perm[nextcard]=perm[j] mod 18 then foundmate:=TRUE:fi:
		od:
	fi:
	nummoves:=nummoves+1:nextcard:=nextcard+1:
od:  
#
RETURN(nummoves);
end:


numhands:=array(1..36);
for i from 1 to 36 do numhands[i]:=0:od:
for i from 1 to 1000 do x:=moves(RandElement(G)):numhands[x]:=numhands[x]+1:od:
#Warning: first invocation of  RandElement  takes a long time.
print(seq(numhands[i],i=1..36));

                                                             v--20 is mode   
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 215, 262, 205, 177, 88,
    23, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

A:=array[0..36, 0..36];

                          A := array[0 .. 36, 0 .. 36]

for m from 0 to 36 do A[0,m]:=0:od: #no unknowns ought to mean m=0, too!
for m from 1 to 36 do A[1,m]:=0:od:A[1,1]:=1:
for n from 2 to 36 do
  for k from 0 to floor(18-n/2) do #actually only need k >= 9-n/2
    m:=36-2*k-n:                   #since  m <= 18 in actual play.
    A[n,m]:=1+(m/n)*A[n-1,m-1]+((n-m)/n/(n-1))*A[n-2,m]
      +((n-m)*(n-m-2)/n/(n-1))*A[n-2,m+2]+ ((n-m)*m/n/(n-1))*A[n-2,m]:
  od:
od:



for i from 0 to 10 do print(seq(A[i,(i mod 2)+2*j],j=0..18));od;