import time
start = time.time()
 
#2x2 square takes 4 steps to reach the end
#so a 20x20 takes 40 steps
#solve mathematically with 40 choose 20
x = y = 1
for i in range(1, 41):
    x = x * i
for i in range(1, 21):
    y = y * i
print x/y/y
 
print time.time()-start

import time
start = time.time()
 
def nextSequence(n):
    #implement rules as laid out in question
    if n%2 == 0:
        return n/2
    else:
        return (n*3)+1
 
#loop through required range
#then loop back down to 1 using nextSequence
chains = [0,1]
for i in range(2,1000000):
    j = i
    x = 0
    while j > 1:
        j = nextSequence(j)
        #check if we have already visited this number
        #if so we already have the chain from this point
        if j < len(chains):
            x += chains[j]
            break
        x += 1
    chains.append(x)
 
print chains.index(max(chains))
 
print time.time()-start

import time
start = time.time()
 
#set the numbers as a string so we can use string methods and operators
numStr = '\
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'
 
#loop through string one 50-digit number at a time and add it to total
x = 0
for i in range(0, 4999, 50):
    x += int(numStr[i:i+49])
 
#convert to string and print first 10 chars
print str(x)[:10]
 
print time.time()-start

import time
start = time.time()
 
def countFactors(n=3, c=2):
    #loop to square root, test with modulus, check for squares
    for i in range(2, int(n**0.5)+1):
        if n%i == 0:
            if i != int(n**0.5)+1:
                c+=2
            else:
                c+=1
    return c
 
def triangleNumbers(i=1):
    #generate triangle numbers with 0.5*n*n1
    while True:
        yield int(0.5*i*(i+1))
        i+=1
 
#loop and a half, generate triangle numbers and count factors
triNums = triangleNumbers()
while True:
    n = triNums.next()
    if countFactors(n) > 500:
        print n
        break
 
print time.time()-start

import time
start = time.time()
 
#build the grid
grid = [[8,02,22,97,38,15,00,40,00,75,04,05,07,78,52,12,50,77,91,8],
[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],
[67,26,20,68,02,62,12,20,95,63,94,39,63,8,40,91,66,49,94,21],
[24,55,58,05,66,73,99,26,97,17,78,78,96,83,14,88,34,89,63,72],
[21,36,23,9,75,00,76,44,20,45,35,14,00,61,33,97,34,31,33,95],
[78,17,53,28,22,75,31,67,15,94,03,80,04,62,16,14,9,53,56,92],
[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],
[04,52,8,83,97,35,99,16,07,97,57,32,16,26,26,79,33,27,98,66],
[88,36,68,87,57,62,20,72,03,46,33,67,46,55,12,32,63,93,53,69],
[04,42,16,73,38,25,39,11,24,94,72,18,8,46,29,32,40,62,76,36],
[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]]
 
highScore = 0
product = 0
 
#use nested loops and offsets to work out products
 
#horizontal
for i in range(17):
    for j in range(20):
        product = grid[j][i] * grid[j][i+1] * grid[j][i+2] * grid[j][i+3]
        if product > highScore:
            highScore = product
 
#vertical
for i in range(20):
    for j in range(17):
        product = grid[j][i] * grid[j+1][i] * grid[j+2][i] * grid[j+3][i]
        if product > highScore:
            highScore = product
 
#left diagonal
for i in range(17):
    for j in range(17):
        product = grid[j][i+3] * grid[j+1][i+2] * grid[j+2][i+1] * grid[j+3][i]
        if product > highScore:
            highScore = product
 
#right diagonal
for i in range(17):
    for j in range(17):
        product = grid[j][i] * grid[j+1][i+1] * grid[j+2][i+2] * grid[j+2][i+2]
        if product > highScore:
            highScore = product
 
print highScore
 
print time.time()-start

import time
start = time.time()
 
#solve with the Sieve of Eratosthenes
target = 2000000
sieve= range(target+1)
for i in range(2, (int(target**0.5)+1)):
    if sieve[i]:
	for j in range(i**2, target+1, i):
            sieve[j] = 0
 
print sum(sieve)-1
 
print time.time()-start

import time
start = time.time()
 
#solve with Euclid's formula
def eulerNine(target):
    for m in range(1,target):
        for n in range(1,m):
            a = m**2-n**2
            b = 2*m*n
            c = m**2+n**2
            if a+b+c == target:
                return a*b*c
print eulerNine(1000)
 
print time.time()-start

import time
start = time.time()
 
def productsByFives(nStr, offset=0, highScore=0):
    if offset+5 > len(nStr):
        #end of string reached
        return highScore
    #use offsets to get product and compare to running highscore
    product = int(nStr[offset])*int(nStr[offset+1])*int(nStr[offset+2])*int(nStr[offset+3])*int(nStr[offset+4])
    if product > highScore:
        highScore = product
    #call function again with offset incremented
    return productsByFives(nStr, offset+1, highScore)
 
#set number as string so we can use string methods and operators
nStr = '7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450'
 
print productsByFives(nStr)
 
print time.time()-start

import time
start = time.time()
 
def isPrime(n):
    #using trial division
    n = abs(int(n))
    if n < 2:
        return False
    if n == 2:
        return True
    if not n & 1:
        return False
    for x in range(3, int(n**0.5)+1, 2):
        if n % x == 0:
            return False
    return True
 
target = 10001
#one var to track primes
#one to increment numbers to test
primes = i = 0
while primes < target:
    i += 1
    if isPrime(i):
        primes += 1
 
print i
 
print time.time()-start

import time
start = time.time()
 
n = 100
print (sum(range(1,n+1))**2) - (sum(x*x for x in range(1,n+1)))
 
print time.time()-start