Description

import numpy as np from numpy import * import math import random import matplotlib.pyplot as plt
In [87]:
Q1
Write a function Print_values with arguments a, b, and c to reflect the following flowchart. Here the purple parallelogram operator on a list [x, y, z] is to compute and print x+y-10z . Try your output with some random a, b, and c values. Report your output when a = 10, b = 5, c = 1.
def Print_values(a,b,c):
temp =[a,b,c] if a>b: if b>c:
temp =[a,b,c] elif a>c:
temp =[a,c,b] else:
temp =[c,a,b] elif b>c: if a>c:
temp =[a,c,b] else:
temp =[c,a,b] else: temp =[c,b,a] return temp[0]+temp[1]-10*temp[2]
In [70]:
a,b,c =10,5,1
print(Print_values(a,b,c))
In [71]: 5
Q2
Given a list with N positive integers. For every element x of the list, find the value of continuous ceiling function defined as F(x) = F(ceil(x/3)) + 2x , where F(1) = 1.
def celing(x): if x ==1: return 1 return celing(math.ceil(x/3))+2*x
In [72]:
N =input(“input a list of positive intergers of x: “).split(‘,’)
In [75]:
res =[] for x in N:
if x.isdigit() and int(x)>0: res.append(celing(int(x))) else:
print(‘please input positive intergers’) break print(‘input:’,N,’ ‘,’output:’,res)
input: [‘1’, ‘3’, ‘7’, ’14’, ’18’, ’22’, ’46’, ’72’, ’89’] output: [1, 7, 21, 43, 53, 67, 141, 215, 271]
Q3
3.1
Given 10 dice each with 6 faces, numbered from 1 to 6. Write a function
Find_number_of_ways to find the number of ways to get sum x, defined as the sum of values on each face when all the dice are thrown.
https://www.geeksforgeeks.org/dice-throw-dp-30/
eg
10次丢出和15的次数 = 第9次丢出和分别为[9，10，11，12，13，14]的总和，同
样的，第9次里丢出和14的次数 =第8次丢出和为[8，9，10，11，12，13]的总
和，以此递推
def Find_number_of_ways(x):
table=[[0]*(x+1) for i in range(11)] if x < 10 or x > 60:
return ‘please input within right range~’
for j in range(1,min(7,x+1)):
table[1][j]=1 for i in range(2,11): for j in range(1,x+1): for k in range(1,min(7,j)): table[i][j]+=table[i-1][j-k] return table[-1][-1]
In [76]:
In [77]: X3 =int(input(“input the sum(range from 10 to 60): “)) print(‘input:’,X3,’ ‘,’output:’,Find_number_of_ways(X3))
input: 32 output: 3801535
3.2
Count the number of ways for any x from 10 to 60, assign the number of ways to a list called Number_of_ways , so which x yields the maximum of Number_of_ways ?
Number_of_ways =[] for i in range(10,61):
Number_of_ways.append(Find_number_of_ways(i)) print(‘The value’,Number_of_ways.index(max(Number_of_ways))+1,’yields the maximum of N
In [78]:
The value 26 yields the maximum of Number of ways
Q4
4.1
Write a function Random_integer to fill an array of N elements by randomly selecting integers from 0 to 10.
In [80]: def Random_integer(N): return np.random.randint(11, size =N) print(‘randomly create a list :’,Random_integer(5))
randomly create a list : [ 6 10 0 7 9]
4.2
Write a function Sum_averages to compute the sum of the average of all subsets of the array. For example, given an array of [1, 2, 3], you Sum_averages function should compute the sum of: average of [1], average of [2], average of [3], average of [1, 2], average of [1, 3], average of [2, 3], and average of [1, 2, 3].
from numpy import *
def Sum_averages(nums):
res = [[]] avg =[] for i in range(len(nums)):
temp = [] for j in range(len(res)):
temp.append(res[j] + [nums[i]]) avg.append(mean(temp[j])) res += temp return sum(avg)
In [81]:
In [82]: nums =list(map(int,input(“input the list of values: “).split(‘,’))) print(‘input:’,nums,’ ‘,’output:’,Sum_averages(nums))
input: [1, 2, 3, 4, 5, 6, 7, 8, 9] output: 2555.0
4.3
Call Sum_averages with N increasing from 1 to 100, assign the output to a list called Total_sum_averages . Plot Total_sum_averages , describe what do you see.
Total_sum_averages =[] for i in range(1,21):
temp =Random_integer(i)
Total_sum_averages.append(Sum_averages(temp)) print(‘the mean of total number of paths from the 20 runs is: ‘,mean(Total_sum_averag
In [86]:
the mean of total number of paths from the 20 runs is: 530279.2800440275
In [101… h =np.linspace(1,21,20) plt.plot(h,Total_sum_averages) plt.ylabel(‘total sum averages’)
Out[101]: Text(0, 0.5, ‘total sum averages’)

5. Path counting
5.1
Create a matrix with N rows and M columns, fill the right-bottom corner and top-left corner cells with 1, and randomly fill the rest of matrix with integer 0 or 1.
In [102… import numpy as np
def create_matrix(N,M):
temp =np.random.randint(0,2,(N,M)) temp[0,0] =1 temp[N-1,M-1] =1 return temp
In [104… mat =list(map(int,input(“create a matrix with N rows and M columns: “).split(‘,’))) ran_mat =create_matrix(mat[0],mat[1]) print(“The new matrix is: “,ran_mat)
The new matrix is:
[[1 1 0 … 0 0 1]
[1 0 1 … 1 0 1]
[1 0 1 … 0 0 1]
…
[1 0 1 … 1 1 0]
[0 0 1 … 1 0 0]
[0 0 1 … 1 1 1]]
5.2
Consider a cell marked with 0 as a blockage or dead-end, and a cell marked with 1 is good to go. Write a function Count_path to count total number of paths to reach the right-bottom corner cell from the top-left corner cell.
def Count_path(i,j,row,col,A): if i ==row or j ==col: return 0
if A[i][j] ==0: return 0
if i ==row-1 and j ==col-1: return 1 return Count_path(i+1, j, row, col, A)+Count_path(i, j+1,row, col, A)
In [106…
5.3
Let N = 10, M = 8, run Count_path for 1000 times, each time the matrix (except the rightbottom corner and top-left corner cells, which remain being 1) is re-filled with integer 0 or 1 randomly, report the mean of total number of paths from the 1000 runs.
paths =[] for i in range(1001): A =create_matrix(10,8)
paths.append(Count_path(0,0,10,8,A)) print(‘the mean of total number of paths from the 1000 runs is: ‘,mean(paths))
In [108…
the mean of total number of paths from the 1000 runs is: 0.3596403596403596