Locks in a grid problem

This is a simulation to an interesting problem about a lock. Which has n key slots. These slots are either vertical or horizontal. To open the lock these slots have to be in a particular configuration. If we put the key in one slot and turn its orientation all the other slots in that particular row and column change their configuration. Is it possible to open this lock for a particular configuration of 16 slots ? If yes then how ?

       

"""The purpose of this is to simulate the problem of locks
where when you turn on the key in the grid all the other
keys in the same row and column also turn. I represent this
by 0 and 1
One can experiment and learn that if the grid is even by even
then we can change any particular bit by"""
n = 4
a=[]
sum = 0
# It prints the whole Grid of Locks
def printMatrix(val):
    for i in range(n):
        print(val[i])

# It flips the entries from 0 to 1 and 1 to 0        
def flip(val):
    if val == 0:
        return 1
    return 0

# It changes the entries in a given row and column 
def ChangeRowChangeColumn(a,i,j):
    a[i][j]=flip(a[i][j])
    for k in range(n):
        a[i][k]=flip(a[i][k])
    for k in range(n):
        a[k][j]=flip(a[k][j])
        
        
for i in range(1,n+1):
    b=[]
    for j in range(1,n+1):
        b.append(1)
    c = b[:]
    sum= sum+n
    a.append(c)
    del b[:]   
printMatrix(a)
flag=1
while(flag == 1):
    row=int(input("Enter what row you want to change ?"))
    col=int(input("Enter what column you want to change?"))
    ChangeRowChangeColumn(a,row,col)
    printMatrix(a)
    flag = int(input("Enter 0 or 1 to continue"))
    
         

       
 

Coupon Collector Problem and Python

Collecting Coupons: This is a famous problem in Mosteller's book.
Coupons in cereal boxes are numbered 1 to 5, and a set of one of each is required for a prize. With one coupon per box, how many boxes on the average are required to make a complete set?

Here is a python program to simulate it

"""We store the outputs in a list a until we get all the coupons
The coupons are generated from the randint function which generates
both the end values
The sum variable keep tracks of number of boxes we have been counting
"""
import random
flag = 1
a=[]
sum = 0
n = 1000 # number of times you want to run the experiment
coupNum = 30 # number of different coupons
for i in range(n):
    count = 0
    a=[]
    flag =1
    while flag:
        x=random.randint(1,coupNum)
        #generates random number between 1 and coupNum inclusive
        count += 1
#        print(x)
        if x not in a:
            a.append(x)
            flag = 1
        elif len(a)==coupNum:
            flag=0
#    print(count)
    sum = sum+count
#    print(a,count,sum)
    del a[:] #delete the list
print(sum/n)

Odds of Craps

Craps

 

The game of craps, played with two dice, is one of America's fastest and most popular gambling games. Calculating the odds associated with it is an instructive exercise. The rules are these. Only totals for the two dice count. The player throws the dice and wins at once if the total for the first throw is 7 or 11, loses at once if it is 2, 3 or 12. Any other throw is called his "point". If the first throw is a point, the player throws the dice repeatedly until he either wins by throwing his point again or loses by throwing 7. What is the player's chance to win.

Python Program to find the final probability
sum = 8/36
a = {4:3/36,5:4/36,6:5/36,8:5/36,9:4/36,10:3/36}
for i in a:
    sum = sum+6*(a[i])**2/(6*a[i]+1)
print(sum)

Prisoner hat problem

This is a problem about a prison where an evil wizard comes and tell the 64 inmates there that tomorrow he will come and put a hat of either black or white color on their head after making them stand in a line. So that they can only see the prisoners in front of them and their hat. He would then start from the inmate at the very end who can see everyone else's hat except for his own. He would ask each of them what color is their hat. They can only answer black or white and no other words. If they are correct he would leave them otherwise he will kill them. They have a whole night to plan their strategy once he puts them in line they cannot communicate.

The problem is to find out a strategy which would allow maximum number of inmates to escape death.

[sourcecode language="python"]
# -*- coding: utf-8 -*-
"""
Created on Wed Nov 16 21:03:20 2016

@author: Sumant
The hats take the input as a STRING of Black and White
Black: Represents even number of black hats
White: Represents odd number of black hats
Black = 1
White = 0
def whatTheySee()
  returns an array which contains the parity of black hats they see in front
  of them

def whatTheySay()
  returns an array which contains what they say

Algorithm: WhatTheySay
  We need to use the array whatTheySee and we return an array which contains
  what each person should say: except for the first entry which corresponds to
  the last number we see. All others are always correct.
  The nth position of the array we return populates itself with the responses
  of previous n-1 responses and add what the parity of black hats the person
  standing at nth position sees.

Input: WWWWBB
Output: WWWWBB

Input: BBBBB
Output: BBBBB

Input: BBBB
Output: WBBB

Input: BWBWBB
Output: WWBWBB
"""

hats="BWBWBB"




def binaryTohat(hat):
    change=[]
    for i in hat:
        if i == 1:
            change.append('B')
        else:
            change.append('W')
    return change
   
def hatTobinary(hat):
    change=[]
    for i in hat:
        if i == 'B':
            change.append(1)
        else:
            change.append(0)
    return change
       
HATS=hatTobinary(hats)

def whatTheySee(hats):

    a=[]
    for j in range(len(hats)):
        sum = 1
        for i in range(j+1,len(hats)):
            sum= sum+HATS[i]
        sum = sum %2
        a.append(sum)
       
    return a

def whatTheySay(hats):
    a=[]
    val=0
    a.append(hats[0])
    for i in range(1,len(hats)):
        val=sum(a)+hats[i]
        a.append(val%2)
    return a      

def stringArraytoString(st):
    a=""
    for i in st:
        a+=i
    return a
print("What wizard see",hats)
print("What they say",stringArraytoString(binaryTohat(whatTheySay(whatTheySee(HATS)))))

[/sourcecode]