# Please do not post homework
# Vikrant, Apr 17 2022, Assignment 22

# ================================================================================
# 0. Code that has been given.
# ================================================================================

read("C22.txt"):

# ================================================================================
# 1. Show that alpha is the golden ratio assuming ai(i)=trunc(i*alpha).
# ================================================================================

(*

# The value of alpha must be unique.
# That is, if ai(i)=trunc(i*alpha)=trunc(i*beta), then alpha=beta.
# Why?
# Let k be the first decimal position in which alpha differs from beta.
# Then trunc(10^k*alpha)<>trunc(10^k*beta).

# TODO.

*)

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# 2. <Skipped>
# ================================================================================

# ================================================================================
# 3. Read and understand LaG(G) and its subroutines.
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# Done.

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# 4. Losing positions.
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LpG:=
    proc(G)
        local os:= LaG(G):
        local i:
        {seq(`if`(os[i]=0,i,NULL),i=1..nops(G))}:
    end:

# ================================================================================
# 5. Average size of losing positions given arc probability.
# ================================================================================

AvNuLP:=
    proc(n,p,K)
        local i:
        evalf(add(nops(LpG(RDG(n,p))),i=1..K)/K):
    end:

(*
randomize(284766820):
for n in [10,50,100] do
    for p in [1/5,2/5,3/5,4/5] do
        print(AvNuLP(n,p,10^5));
    od:
od:
*)