# OK to post homework
# RDB, 2022-03-04, HW12

# 1.

with(ListTools):

DecDigits := proc(a, K)
    Reverse(convert(op(1, evalf(a, K)), base, 10)):
end:

# 2.

WrongDigits := proc(a, b, K)
    local A, B:
    A := DecDigits(a, K):
    B := DecDigits(b, K):
    select(k -> A[k] <> B[k], [seq(1..K)]):
end:

read "C15.txt":

# 3.

# There is no mathematician with a PhD with the initials R. N. according to the
# Mathematics Genealogy Project.

# So, whoever R.D. North is, they probably didn't get a PhD in math!

# 4.

Archimedes := proc(k, N)
    [k * 2^N * sin(Pi / k / 2^N), k * 2^N * tan(Pi / k / 2^N)]:
end:

# 5.

# We want k * ATAN(a) + ATAN(b) = 1, or b = tan(1 - k * ATAN(a)).

# We want the b which is closest to 0. The exact values of k which give zero
# are

#   1 - k * ATAN(a) = n * pi

# for any integer n. Thus, we want

#   k = (1 - n * pi) / ATAN(a).

# This is not going to be an integer, but we can take the integer nearest it.
# There's some theorem that says the fractional parts of n * x are
# equidistributed when x is irrational, so I suspect that for almost all a
# there is no "best" candidate for n.