def recursive_paths(dimensions, path_history=""):
"""
calculates the number of paths through an m x n grid
dimensions (list): specifies an m x n grid as [m, n]
returns (int): number of paths
"""
# logging
stub = Stub()
stub.start()
# base case
if dimensions == [0, 0]:
stub.msg(path_history, "path found")
return 1
# path and history initialization
paths = 0
path_history_r = path_history_d = path_history
# recursion
if dimensions[0] > 0:
path_history_r += "r"
paths += recursive_paths([dimensions[0]-1, dimensions[1]], path_history_r)
if dimensions[1] > 0:
path_history_d += "d"
paths += recursive_paths([dimensions[0], dimensions[1]-1], path_history_d)
return paths
def longest_collatz_sequence(limit):
"""
exhaustively finds the longest collatz sequence from starting numbers
up to a certain limit, exclusive
limit (int): limit, exclusive
returns (int): starting number
"""
stub = Stub()
stub.start()
max_steps = 0
max_start_num = 1
collatz_branches = {}
for num in range(1, limit):
next_num = next_collatz_number
if next_num in collatz_branches:
collatz_sequence = [num] + collatz_branches[next_num]
else:
collatz_sequence = get_collatz_sequence(num)
steps = len(collatz_sequence)
if steps > max_steps:
max_steps = steps
max_start_num = num
stub.msg(str(max_steps) + " steps", max_start_num)
collatz_branches[num] = collatz_sequence
return max_start_num
def largest_palindrome_product(minimum=100, maximum=999):
"""
exhaustively finds the largest palindromic product in a range
minimum, maximum (int): defines range, inclusive
returns (int): solution
"""
stub = Stub()
#stub.start()
candidate_solution = None
palindrome_products = []
candidate_factors = [None, None]
for i in range(maximum, minimum-1, -1):
for j in range(i, minimum-1, -1):
product = i*j
if is_palindrome(product):
palindrome_products.append(product)
if product > candidate_solution:
candidate_solution = product
candidate_factors = [i, j]
stub.msg(candidate_factors, "factors")
return candidate_solution
def get_fib_with_digits(num_digits):
"""
finds the first Fibonacci number with a certain amount of digits
"""
stub = Stub()
stub.start()
counter = 0
for num in gen_fibonacci():
if count_digits(num) >= num_digits:
return counter
if counter % 1000 == 0:
stub.msg(count_digits(num), counter)
counter += 1
def sum_square_difference(limit):
"""
finds the difference of the square of sums and sum of squares for
numbers 1 through some number
limit (int): limit, inclusive
returns (int): difference
"""
stub = Stub()
#stub.start()
sum_of_squares = sum([num**2 for num in range(limit+1)])
square_of_sums = sum([num for num in range(limit+1)])**2
stub.msg(sum_of_squares, "sum")
stub.msg(square_of_sums, "square")
return square_of_sums - sum_of_squares
def special_pythagorean_triplet():
"""
finds the special pythagorean triplet that also sums to 1000
returns (int): product of triplet
"""
stub = Stub()
# stub.start()
# generate power set of numbers that sum to 1000
counter = 0
for triplet in gen_triplets_that_sum_to(1000):
if is_pythagorean(*triplet):
stub.msg(triplet, counter)
return product(triplet)
counter += 1
raise ValueError("unable to find special pythagorean triplet after " + str(counter) + " trials.")
def largest_prime_factor(num):
"""
finds the largest prime factor of a number
num (int): some number
returns (int): solution
"""
stub = Stub()
#stub.start()
j = 0
for i in range(2, int(num**0.5)):
j += 1
while num % i == 0:
stub.msg(i, j)
num //= i
if num == 1 or num == i:
return i
def sum_of_amicable_numbers(limit):
"""
finds the sum of "amicable numbers" up to some limit
limit (int): limit, exclusive > 1
returns (int): solution
"""
stub = Stub()
stub.start()
stack = []
for num in range(2, limit):
if num in stack:
continue
elif is_amicable(num):
stack.append(num)
stack.append(sum_of_proper_divisors(num))
stub.msg(stack[-2:])
return sum(stack)
def largest_product_in_a_grid(grid, chunk_len):
"""
finds the largest consecutive product in a grid (horiz, vert, diag)
grid (list): 2d list or array representing grid dimensions and values
chunk_len (int): length of consecutive numbers to check
returns (int): solution
"""
stub = Stub()
#stub.start()
grid_rows = len(grid)
grid_cols = len(grid[0])
largest = 0
# check N-S, W-E, NW-SE products
for i in range(grid_rows - chunk_len):
for j in range(grid_cols - chunk_len):
n_s = [grid[i+n][j] for n in range(chunk_len)]
w_e = grid[i][j:j+chunk_len]
nw_se = [grid[i+n][j+n] for n in range(chunk_len)]
for chunk in [n_s, w_e, nw_se]:
if product(chunk) > largest:
stub.msg(chunk)
largest = product(chunk)
# check SW-NE products
for i in range(chunk_len-1, grid_rows):
for j in range(grid_cols - chunk_len):
sw_ne = [grid[i-n][j+n] for n in range(chunk_len)]
if product(sw_ne) > largest:
stub.msg(sw_ne)
largest = product(sw_ne)
return largest
def large_sum(large_numbers, first_digits):
"""
finds an estimated sum of several large numbers
large_numbers (list): unordered list of large numbers
first_digits (int): number of leading digits to calculate
returns (int or long): solution
"""
stub = Stub()
#stub.start()
max_digits = len(str(large[0]))
places_to_check = first_digits + int(log10(len(large_numbers)*10))
stub.msg(places_to_check, "places to check")
sum_estimate = 0
for place in range(max_digits, max_digits-places_to_check, -1):
temp_sum = sum([int(str(num)[max_digits-place]) for num in large_numbers])
stub.msg((str(temp_sum) + "*10^" + str(place)), "intermediate sum")
sum_estimate += temp_sum * 10**place
stub.msg(sum_estimate, "final sum")
return sum_estimate