# OK to post homework
# James Betti, 2 Mar 2025, Assignment 11

with(combinat):

CheckMul := proc(n) local A,i,j;
    A := Bn(n);
    A := [seq([seq(sqrt(n)*A[i][j],j=1..nops(A[i]))],i=1..nops(A))];
    A := Mul(An(n),Bn(n))-A;
    for i to nops(A) do
        for j to nops(A[i]) do
            if A[i][j]<>0 then RETURN(false) fi;
        od;
    od;
    true;
end:

CheckMul(10);                           # Verified up to n=10

IndNu := proc(G) local r,i;
    r := -1;
    for i to G[1] do
        if MinMaxNNpablo(G,i)=0 then
            r := max(i,r);
        fi;
    od;
    r;
end:

for i from 4 to 15 do
    printf("IndNu(Gnd(%d,2)):",i);
    IndNu(Gnd(i,2));
    printf("IndNu(Gnd(%d,3)):",i);
    IndNu(Gnd(i,3));
od;

# ====================RESULTS====================
# IndNu(Gnd( 4,2)): 1         IndNu(Gnd( 4,3)): 1
# IndNu(Gnd( 5,2)): 1         IndNu(Gnd( 5,3)): 1
# IndNu(Gnd( 6,2)): 2         IndNu(Gnd( 6,3)): 1
# IndNu(Gnd( 7,2)): 2         IndNu(Gnd( 7,3)): 1
# IndNu(Gnd( 8,2)): 2         IndNu(Gnd( 8,3)): 2
# IndNu(Gnd( 9,2)): 3         IndNu(Gnd( 9,3)): 2
# IndNu(Gnd(10,2)): 3         IndNu(Gnd(10,3)): 2
# IndNu(Gnd(11,2)): 3         IndNu(Gnd(11,3)): 2
# IndNu(Gnd(12,2)): 4         IndNu(Gnd(12,3)): 3
# IndNu(Gnd(13,2)): 4         IndNu(Gnd(13,3)): 3
# IndNu(Gnd(14,2)): 4         IndNu(Gnd(14,3)): 3
# IndNu(Gnd(15,2)): 5         IndNu(Gnd(15,3)): 3

UpperBoundMinMaxNN := proc(G,r,K) local m,_,H,i;
    m := infinity;
    for _ to K do
        H := randcomb({seq(i,i=1..G[1])},r);
        m := min(m,MaxNN(G,H));
    od;
    m;
end:

UpperBoundMinMaxNN(Ck(8),129,5000);     # 6