def pe202():
"""
| \| \| \| \| \| \| Expand triangle mirrors infinitely, then every
+--+--+--+--+--+--+- possible path is a C-C line from bottomleft to
C|\B|\A|\C|\B|\A|\C|\ somewhere upright and doesn't cross any vertex.
| \| \| \| \| \| \|
+--+--+--+--+--+--+- From the table can easity see that C-C lines are
B|\A|\C|\B|\A|\C|\B|\ symmetric and lie on y = x + 3k. Without loss of
| \| \| \| \| \| \| generality, we just need to count all (a, b) with
+--+--+--+--+--+--+- a > b.
A|\C|\B|\A|\C|\B|\A|\
| \| \| \| \| \| \| Also, from (0, 0) to (a, b), the amount of cross
+--+--+--+--+--+--+- points with horizontal, vertical and diagonal
C B A C B A C lines is (a-1) + (b-1) + (a+b-1) = 2a + 2b - 3.
Therefore to bounce N times, we need to find all (a, b) that:
(1) a + b = (N + 3) / 2
(2) gcd(a, b) = 1
(3) a - b = 3k
Recording to reference articles PE202, the number of solutions of (1), (2)
and (3) is:
sum m(d) * f(n/d) for all d satisfying d|n
where m(x) if mobius function, f(x) = floor(x/2) - floor(x/3)
"""
def f(x):
return x / 2 - x / 3
N = 12017639147
N = (N + 3) / 2 # 6008819575
# 6008819575 = 5*5 * 11 * 17 * 23 * 29 * 41 * 47
pfactors = (5, 11, 17, 23, 29, 41, 47)
count = f(N)
for i in range(1, len(pfactors)+1):
for pcomb in pec.combinations(pfactors, i):
d = pef.cprod(pcomb)
if i % 2:
count -= f(N / d)
else:
count += f(N / d)
# answer: 1209002624
return count * 2
def pe184(r=105):
"""
Find all points in first quardrant, then group them by slope (a.k.a. lines).
Count how many points between each line and y-axis.
There 5 different types of triangles. Count them out.
"""
from fractions import gcd
def count(p, r):
xp, yp = p
xmax = int(math.sqrt(1. * r*r / (xp*xp + yp*yp)) * xp)
yk = 1. * yp * xmax / xp
if xmax*xmax + yk*yk < r*r:
xmax += 1
c = 0
for x in range(1, xmax):
ymax = int(math.sqrt(r*r - x*x))
if x*x + ymax*ymax == r*r:
ymax -= 1
ymin = int(1. * x * yp / xp) + 1
c += ymax - ymin + 1
return c
nq1, nline = 0, {}
for x in range(1, r):
for y in range(1, int(math.sqrt(r*r - x*x)) + 1):
if x*x + y*y < r*r:
nq1 += 1
g = gcd(x, y)
xr = x / g
yr = y / g
if (xr, yr) in nline:
nline[(xr, yr)] += 1
else:
nline[(xr, yr)] = 1
# a(*4): one apex on y+, one apex on x+
# b(*8): one apex on y+, one apex in 1st quardrant
# c(*4): one apex on y+, one apex in 2nd quardrant, one apex in 3rd quardrant
# d(*4): two apexes in first quardrant
# e(*2): three apexes in different quardrants
a = (r-1) * (r-1) * nq1
b = c = d = e = 0
nty, ntx = {}, {}
for (x, y), n in nline.iteritems():
nty[(x, y)] = count((x, y), r)
ntx[(x, y)] = nq1 - n - nty[(x, y)]
b += (r-1) * n * nty[(x, y)]
c += (r-1) * n * nq1
e += 2 * n * n * ntx[(x, y)]
for (x1, y1), (x2, y2) in pec.combinations(nline.keys(), 2):
if 1.*y2/x2 > 1.*y1/x1:
x1, y1, x2, y2 = x2, y2, x1, y1
coeff = nline[(x1, y1)] * nline[(x2, y2)]
n = nty[(x2, y2)] - nty[(x1, y1)] - nline[(x1, y1)]
d += coeff * n
e += 2 * coeff * (n + nline[(x2, y2)] + 2 * ntx[(x2, y2)])
# answer: 1725323624056
return a*4 + b*8 + c*4 + d*4 + e*2