problems_11_to_20.py

from math import factorial
# Project Euler problems
# Problem 11
grid = [
    "08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08",
    "49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00",
    "81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65",
    "52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91",
    "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"
    ]

def cell(x, y):
    return int(grid[y-1][(x-1)*3:(x-1)*3+2])

def all_combinations():
    for y in range(1, 21):
        for x in range(1, 18):
            yield [cell(x, y), cell(x+1, y), cell(x+2, y), cell(x+3, y)]

    for x in range(1, 21):
        for y in range(1, 18):
            yield [cell(x, y), cell(x, y+1), cell(x, y+2), cell(x, y+3)]

    for x in range(1, 18):
        for y in range(1, 18):
            yield [cell(x, y), cell(x+1, y+1), cell(x+2, y+2), cell(x+3, y+3)]

    for x in range(4, 21):
        for y in range(1, 18):
            yield [cell(x, y), cell(x-1, y+1), cell(x-2, y+2), cell(x-3, y+3)]

def product(values):
    result = 1
    for value in values:
        result *= value
    return result

def all_products():
    for four_numbers in all_combinations():
        yield product(four_numbers)
print(max(all_products()))

# Problem 12
# What is the value of the first triangle number to have over five hundred divisors?

def triangle_number(n):
    return int(n*(n+1)/2)

def divisors(n): # doesn't work for squares
    divisors = 0
    for i in range(1, int(n**0.5)+1):
        if n % i == 0:
            divisors += 2
    return divisors

n = 1
while divisors(triangle_number(n)) < 500:
    n += 1

print(triangle_number(n))

# Problem 13
# Work out the first ten digits of the sum of the following one-hundred 50-digit numbers.

numbers = [37107287533902102798797998220837590246510135740250,
46376937677490009712648124896970078050417018260538,
74324986199524741059474233309513058123726617309629,
91942213363574161572522430563301811072406154908250,
23067588207539346171171980310421047513778063246676,
89261670696623633820136378418383684178734361726757,
28112879812849979408065481931592621691275889832738,
44274228917432520321923589422876796487670272189318,
47451445736001306439091167216856844588711603153276,
70386486105843025439939619828917593665686757934951,
62176457141856560629502157223196586755079324193331,
64906352462741904929101432445813822663347944758178,
92575867718337217661963751590579239728245598838407,
58203565325359399008402633568948830189458628227828,
80181199384826282014278194139940567587151170094390,
35398664372827112653829987240784473053190104293586,
86515506006295864861532075273371959191420517255829,
71693888707715466499115593487603532921714970056938,
54370070576826684624621495650076471787294438377604,
53282654108756828443191190634694037855217779295145,
36123272525000296071075082563815656710885258350721,
45876576172410976447339110607218265236877223636045,
17423706905851860660448207621209813287860733969412,
81142660418086830619328460811191061556940512689692,
51934325451728388641918047049293215058642563049483,
62467221648435076201727918039944693004732956340691,
15732444386908125794514089057706229429197107928209,
55037687525678773091862540744969844508330393682126,
18336384825330154686196124348767681297534375946515,
80386287592878490201521685554828717201219257766954,
78182833757993103614740356856449095527097864797581,
16726320100436897842553539920931837441497806860984,
48403098129077791799088218795327364475675590848030,
87086987551392711854517078544161852424320693150332,
59959406895756536782107074926966537676326235447210,
69793950679652694742597709739166693763042633987085,
41052684708299085211399427365734116182760315001271,
65378607361501080857009149939512557028198746004375,
35829035317434717326932123578154982629742552737307,
94953759765105305946966067683156574377167401875275,
88902802571733229619176668713819931811048770190271,
25267680276078003013678680992525463401061632866526,
36270218540497705585629946580636237993140746255962,
24074486908231174977792365466257246923322810917141,
91430288197103288597806669760892938638285025333403,
34413065578016127815921815005561868836468420090470,
23053081172816430487623791969842487255036638784583,
11487696932154902810424020138335124462181441773470,
63783299490636259666498587618221225225512486764533,
67720186971698544312419572409913959008952310058822,
95548255300263520781532296796249481641953868218774,
76085327132285723110424803456124867697064507995236,
37774242535411291684276865538926205024910326572967,
23701913275725675285653248258265463092207058596522,
29798860272258331913126375147341994889534765745501,
18495701454879288984856827726077713721403798879715,
38298203783031473527721580348144513491373226651381,
34829543829199918180278916522431027392251122869539,
40957953066405232632538044100059654939159879593635,
29746152185502371307642255121183693803580388584903,
41698116222072977186158236678424689157993532961922,
62467957194401269043877107275048102390895523597457,
23189706772547915061505504953922979530901129967519,
86188088225875314529584099251203829009407770775672,
11306739708304724483816533873502340845647058077308,
82959174767140363198008187129011875491310547126581,
97623331044818386269515456334926366572897563400500,
42846280183517070527831839425882145521227251250327,
55121603546981200581762165212827652751691296897789,
32238195734329339946437501907836945765883352399886,
75506164965184775180738168837861091527357929701337,
62177842752192623401942399639168044983993173312731,
32924185707147349566916674687634660915035914677504,
99518671430235219628894890102423325116913619626622,
73267460800591547471830798392868535206946944540724,
76841822524674417161514036427982273348055556214818,
97142617910342598647204516893989422179826088076852,
87783646182799346313767754307809363333018982642090,
10848802521674670883215120185883543223812876952786,
71329612474782464538636993009049310363619763878039,
62184073572399794223406235393808339651327408011116,
66627891981488087797941876876144230030984490851411,
60661826293682836764744779239180335110989069790714,
85786944089552990653640447425576083659976645795096,
66024396409905389607120198219976047599490197230297,
64913982680032973156037120041377903785566085089252,
16730939319872750275468906903707539413042652315011,
94809377245048795150954100921645863754710598436791,
78639167021187492431995700641917969777599028300699,
15368713711936614952811305876380278410754449733078,
40789923115535562561142322423255033685442488917353,
44889911501440648020369068063960672322193204149535,
41503128880339536053299340368006977710650566631954,
81234880673210146739058568557934581403627822703280,
82616570773948327592232845941706525094512325230608,
22918802058777319719839450180888072429661980811197,
77158542502016545090413245809786882778948721859617,
72107838435069186155435662884062257473692284509516,
20849603980134001723930671666823555245252804609722,
53503534226472524250874054075591789781264330331690]

print(str(sum(numbers))[:10])

# Problem 14
# Which starting number, under one million, produces the longest chain?

def collatz(n):
    if n % 2 == 0:
        return int(n/2)
    else:
        return 3*n+1
    
def collatz_chain(n):
    chain = [n]
    while n != 1:
        n = collatz(n)
        chain.append(n)
    return chain

longest_chain = 0
longest_chain_start = 0
for i in range(1, 1000000):
    if len(collatz_chain(i)) > longest_chain:
        longest_chain = len(collatz_chain(i))
        longest_chain_start = i

print(longest_chain_start)

# Problem 15
# How many such routes are there through a 20×20 grid?

# This is a combinatorics problem. We have to choose 20 down and 20 right moves
# from 40 total moves. The number of ways to do this is 40!/(20!20!).

print(int(factorial(40)/(factorial(20)*factorial(20))))

# Problem 16
# What is the sum of the digits of the number 2^1000?

print(sum([int(i) for i in str(2**1000)]))

# Problem 17

# If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there
# are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.
# If all the numbers from 1 to 1000 (one thousand) inclusive were written out in words,
# how many letters would be used?

# We can use the num2words package to convert numbers to words.
# We can then remove all spaces and hyphens, and count the number of letters.
from num2words import num2words
list_of_words = [num2words(i) for i in range(1, 1001)]
list_of_words = [i.replace(" ", "") for i in list_of_words]
list_of_words = [i.replace("-", "") for i in list_of_words]
print(sum([len(i) for i in list_of_words]))

# Problem 18
# Find the maximum total from top to bottom of the triangle below:
triangle = [
'75',
'95 64',
'17 47 82',
'18 35 87 10',
'20 04 82 47 65',
'19 01 23 75 03 34',
'88 02 77 73 07 63 67',
'99 65 04 28 06 16 70 92',
'41 41 26 56 83 40 80 70 33',
'41 48 72 33 47 32 37 16 94 29',
'53 71 44 65 25 43 91 52 97 51 14',
'70 11 33 28 77 73 17 78 39 68 17 57',
'91 71 52 38 17 14 91 43 58 50 27 29 48',
'63 66 04 68 89 53 67 30 73 16 69 87 40 31',
'04 62 98 27 23 09 70 98 73 93 38 53 60 04 23']

# We can convert the triangle to a list of lists of integers.
triangle = [i.split(" ") for i in triangle]
triangle = [[int(j) for j in i] for i in triangle]

# We can then use dynamic programming to find the maximum path.
# We start at the bottom and work our way up.
# We add the maximum of the two numbers below to the current number.
# We can then remove the last row and repeat the process.

while len(triangle) > 1:
    for i in range(len(triangle[-2])):
        triangle[-2][i] += max(triangle[-1][i], triangle[-1][i+1])
    triangle.pop()

print(triangle[0][0])

# Problem 19
# How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)?

# We can use the calendar module to find the day of the week for a given date.
import calendar
sundays = 0
for year in range(1901, 2001):
    for month in range(1, 13):
        if calendar.weekday(year, month, 1) == 6:
            sundays += 1
print(sundays)

# Problem 20
# find the sum of the digits in the number 100!

print(sum([int(i) for i in str(factorial(100))]))