1. Q1
Write a program characters_triangle.py that gets a strictly positive integer N as input and outputs a triangle of height N, following this kind of interaction:
def running_time(f):
def g(*args):
from time import time
before = time()
f(*args)
after = time()
print(f'It took {after - before} seconds to execute the function.')
return g
while True:
try:
N = int(input('Enter strictly positive number: '))
if N < 1:
raise ValueError
break
except ValueError:
print('The input is illegal. Please input again.')
array = [[0] * (N * 2 - 1) for _ in range(N)]
#根據輸入的N創建所有element為0的矩陣
@running_time
def characters_triangle(N):
center = N - 1
#所有row的center不變
array[0][center] = 65
#初始化第一行的center值，是字母A的ord值
if N != 1:
for i in range(1, N):
last_center = array[i - 1][center]
array[i][center] = last_center + i + 1
#新一行的中心值等于上一行的中心值 + 行號 + 1
end = array[i][center] - 1
#從中心值向兩側賦值
move = 1
#每次移動1位
while end > last_center:
#移動到ord值剛好比上一行中心值大1為止
array[i][center - move] = end
array[i][center + move] = end
#中心值兩側對稱賦值
end -= 1
move += 1
for j in range(2 * N - 1):
#如果有element的值大于90（Z的ord值），則減去26，重新回到A-Z的ord范圍中
while array[i][j] > 90:
#注意要使用循環減去26，一旦N很大，可能超出多個26
array[i][j] -= 26
return array
def print_characters():
for i in range(N):
for j in range(2 * N - 1):
if array[i][j] == 0:
print(' ', end = '')
else:
print(chr(array[i][j]), end = '')
print()
characters_triangle(N)
print_characters()
2. Q2
Write a program pascal_triangle.py that prompts the user for a number N and prints out the first N + 1 lines of Pascal triangle, making sure the numbers are nicely aligned, following this kind of interaction.
def running_time(f):
def g(*args):
from time import time
before = time()
f(*args)
after = time()
print(f'It took {after - before} seconds to execute the function.')
return g
while True:
try:
N = int(input('Enter a nonnegative integer: '))
if N < 0:
raise ValueError
break
except ValueError:
print('The input is illegal. Please input again.')
array = [[0] * (N * 2 + 3) for _ in range(N + 1)]
#根據輸入的N創建所有element為0的矩陣，在原矩陣兩側各加一列0，便于boundary計算
array[0][N + 1] = 1
#確定初始值，第一行中間值為1
@running_time
def pascal():
if N > 0:
for i in range(1, N + 1):
for j in range(1, N * 2 + 2):
array[i][j] = array[i - 1][j - 1] + array[i - 1][j + 1]
#新的一行每個元素等于上一行左上和右上兩個數字的加和
def print_characters():
space = len(str(max(array[-1])))
#為了保持輸出格式，確定最長數字的長度
for i in range(N + 1):
for j in range(1, 2 * N + 3):
e = array[i][j]
if e == 0:
print(' ' * space, sep = '', end = '')
else:
print(' ' * (space - len(str(e))), e, sep = '', end = '')
print()
pascal()
print_characters()
3. Q3
Write a program plane_encoding.py that implements a function encode(a, b) and a function decode(n) for the one-to-one mapping from the set of pairs of integers onto the set of natural numbers, that can be graphically described as follows:
def encode(x, y):
#每一層x開始的數都是(level * 2 - 1) ** 2，坐標都是[level, 1 - level]
#再根據x, y坐標的情況判斷處于該層正方形不同邊上位置的數字是多少
level = max(abs(x), abs(y))
start = (level * 2 - 1) ** 2
start_position = [level, 1 - level]
if x == level and y != -level:
#該層正方形右邊
return start + (y - start_position[1])
elif y == level:
#該層正方形上邊
return start + (2 * level - 1) + (level - x)
elif x == -level:
#該層正方形左邊
return start + (2 * level - 1) + (2 * level) + (level - y)
elif y == -level:
#該層正方形下邊
return start + (2 * level - 1) + (2 * level) + (2 * level) + (x + level)
def decode(n):
if n == 0:
return (0, 0)
level = 1
while (2 * level - 1) ** 2 <= n:
level += 1
level -= 1
start = (level * 2 - 1) ** 2
start_position = [level, 1 - level]
if n <= (start + level * 2 - 1):
#該層正方形右邊
return (level, n - start + start_position[1])
elif n <= (start + level * 2 - 1) + level * 2:
#該層正方形上邊
return (level - (n - (start + 2 * level - 1)), level)
elif n <= (start + level * 2 - 1) + level * 2 + level * 2:
#該層正方形左邊
return (-level, level - (n - (start + 2 * level - 1 + 2 * level)))
else:
return (-level + (n - (start + 2 * level - 1 + 2 * level + 2 * level)), -level)
4. Q4
Given a positive integer n, a magic square of order n is a matrix of size n×n that stores all numbers from1 up to n2 and such that the sum of the n rows, the sum of the n columns, and the sum of the two diagonals is constant, hence equal to n(n2 + 1)/2. The function print_square(square) prints a list of lists that represents a square, and the function is_magic_square(square) checks whether a list of lists is a magic square. For instance:
def print_square(square):
length = len(str(max(max(square))))
for row in square:
for i in range(len(row)):
if i < len(row) - 1:
print(' ' * (length - len(str(row[i]))), row[i], end = ' ')
else:
print(' ' * (length - len(str(row[i]))), row[i])
def is_magic_square(square):
n = len(square)
total = int(n * (n ** 2 + 1) / 2)
for row in square:
if sum(row) != total:
return False
#每一行的和是否成立
total_column = [0] * n
total_diagnal = [0] * 2
for i in range(n):
for j in range(n):
total_column[j] += square[i][j]
#每一列的和
if i == j:
total_diagnal[0] += square[i][j]
#左上右下對角線的和
if i + j == n - 1:
total_diagnal[1] += square[i][j]
#左下右上對角線的和
for e in total_column:
if e != total:
return False
for e in total_diagnal:
if e!= total:
return False
return True
