22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80
24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50
32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70
67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21
24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72
21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95
78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92
16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57
86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58
19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40
04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66
88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69
04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36
20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16
20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54
01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48'''
grid = []
[grid.append(line.split(' ')) for line in nums.split('\n')]
high = 0
for r in range(3, 17):
for c in range(3, 17):
high = max(high, prod([int(grid[r - i][c]) for i in range(0, 4)])) #up
high = max(high, prod([int(grid[r + i][c]) for i in range(0, 4)])) #down
high = max(high, prod([int(grid[r][c - i]) for i in range(0, 4)])) #left
high = max(high, prod([int(grid[r][c + i]) for i in range(0, 4)])) #right
high = max(high, prod([int(grid[r - i][c - i]) for i in range(0, 4)])) #upleft
high = max(high, prod([int(grid[r - i][c + i]) for i in range(0, 4)])) #upright
high = max(high, prod([int(grid[r + i][c - i]) for i in range(0, 4)])) #downleft
high = max(high, prod([int(grid[r + i][c + i]) for i in range(0, 4)])) #downright
print(high)
16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57
86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58
19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40
04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66
88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69
04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36
20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16
20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54
01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48'''
grid = []
[grid.append(line.split(' ')) for line in nums.split('\n')]
high = 0
for r in range(3, 17):
for c in range(3, 17):
high = max(high, prod([int(grid[r - i][c]) for i in range(0, 4)])) #up
high = max(high,
prod([int(grid[r + i][c]) for i in range(0, 4)])) #down
high = max(high,
prod([int(grid[r][c - i]) for i in range(0, 4)])) #left
high = max(high,
prod([int(grid[r][c + i]) for i in range(0, 4)])) #right
high = max(high, prod([int(grid[r - i][c - i])
for i in range(0, 4)])) #upleft
high = max(high, prod([int(grid[r - i][c + i])
for i in range(0, 4)])) #upright
high = max(high, prod([int(grid[r + i][c - i])
for i in range(0, 4)])) #downleft
high = max(high, prod([int(grid[r + i][c + i])
for i in range(0, 4)])) #downright
print(high)
#===============================================================================
# If dn represents the nth digit of the fractional part, find the value of the following expression.
#
# d1 d10 d100 d1000 d10000 d100000 d1000000
#===============================================================================
from Common import prod
intConcat = ''
n = 0
while len(intConcat) <= 1000000:
intConcat += str(n)
n += 1
print(prod([int(intConcat[10**i]) for i in range(7)]))
#===============================================================================
# If dn represents the nth digit of the fractional part, find the value of the following expression.
#
# d1 d10 d100 d1000 d10000 d100000 d1000000
#===============================================================================
from Common import prod
intConcat = ''
n = 0
while len(intConcat) <= 1000000:
intConcat += str(n)
n += 1
print(prod([int(intConcat[10 ** i]) for i in range(7)]))
#===============================================================================
# Considering quadratics of the form:
#
# n^2 + an + b, where |a| < 1000 and |b| < 1000
#
# Find the product of the coefficients, a and b, for the quadratic expression that produces the maximum number of primes for consecutive values of n,
# starting with n = 0.
#===============================================================================
from Common import isPrime, prod
highStreak = 0
coefficients = []
for a in range(-999, 1000):
for b in range(-999, 1000):
count = 0
n = 0
while isPrime(n ** 2 + a * n + b):
count += 1
n += 1
if count > highStreak:
highStreak = count
coefficients = [a, b]
print(prod(coefficients))
#
# There are exactly four non-trivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.
#
# If the product of these four fractions is given in its lowest common terms, find the value of the denominator.
#===============================================================================
from Common import prod, gcd
class Fraction:
def __init__(self, num, den):
self.num = num
self.den = den
def __eq__(self, other):
return self.num * other.den == other.num * self.den
specials = []
for d in range(11, 100):
for n in range(11, d):
if n == d or (n % 10 == 0 and d % 10 == 0):
continue
for c in str(n):
if c in str(d):
frac = Fraction(n, d)
canc = Fraction(int(str(n).replace(c, '', 1)), int(str(d).replace(c, '', 1)))
if frac == canc:
specials.append(frac)
num = prod([f.num for f in specials])
den = prod([f.den for f in specials])
print(den // gcd(num, den))
#===============================================================================
# Find the greatest product of five consecutive digits in the 1000-digit number.
#===============================================================================
from Common import prod
n = '''73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450'''
n = n.replace('\n', '')
print(max([(prod([int(n[d]) for d in range(i, i + 5)])) for i in range(996)]))
#
# If the product of these four fractions is given in its lowest common terms, find the value of the denominator.
#===============================================================================
from Common import prod, gcd
class Fraction:
def __init__(self, num, den):
self.num = num
self.den = den
def __eq__(self, other):
return self.num * other.den == other.num * self.den
specials = []
for d in range(11, 100):
for n in range(11, d):
if n == d or (n % 10 == 0 and d % 10 == 0):
continue
for c in str(n):
if c in str(d):
frac = Fraction(n, d)
canc = Fraction(int(str(n).replace(c, '', 1)),
int(str(d).replace(c, '', 1)))
if frac == canc:
specials.append(frac)
num = prod([f.num for f in specials])
den = prod([f.den for f in specials])
print(den // gcd(num, den))
