Bridge crossing puzzle - algorithm

Question

Four men have to cross a bridge at night.Any party who crosses, either one or two men, must carry the flashlight with them. The flashlight must be walked back and forth; it cannot be thrown, etc. Each man walks at a different speed. One takes 1 minute to cross, another 2 minutes, another 5, and the last 10 minutes. If two men cross together, they must walk at the slower man's pace. There are no tricks--the men all start on the same side, the flashlight cannot shine a long distance, no one can be carried, etc.
And the question is What's the fastest they can all get across. I am basically looking for some generalized approach to these kind of problem. I was told by my friend, that this can be solved by Fibonacci series, but the solution does not work for all.
Please note this is not a home work.

Answer 1

There is an entire PDF (alternate link) that solves the general case of this problem (in a formal proof).

Answer 2

17 minutes - this is a classic MS question.
1,2 => 2 minutes passed.
1 retuns => 3 minutes passed.
5,10 => 13 minutes passed.
2 returns => 15 minutes passed.
1,2 => 17 minute passed.
In general the largest problem / slowest people should always be put together, and sufficient trips of the fastest made to be able to bring the light back each time without using a slow resource.

Answer 3

I would solve this problem by placing a fake job ad on Dice.com, and then asking this question in the interviews until someone gets it right.

Answer 4

As per Wikipedia
The puzzle is known to have appeared as early as 1981, in the book Super Strategies For Puzzles and Games. In this version of the puzzle, A, B, C and D take 5, 10, 20, and 25 minutes, respectively, to cross, and the time limit is 60 minutes
This question was however popularized after its appearance in the book "How Would You Move Mount Fuji?"
the question can be generalized for N people with varying individual time taken to cross the bridge.
The below program works for a generic N no of people and their times.
class Program
{
public static int TotalTime(List<int> band, int n)
{
if (n < 3)
{
return band[n - 1];
}
else if (n == 3)
{
return band[0] + band[1] + band[2];
}
else
{
int temp1 = band[n - 1] + band[0] + band[n - 2] + band[0];
int temp2 = band[1] + band[0] + band[n - 1] + band[1];
if (temp1 < temp2)
{
return temp1 + TotalTime(band, n - 2);
}
else if (temp2 < temp1)
{
return temp2 + TotalTime(band, n - 2);
}
else
{
return temp2 + TotalTime(band, n - 2);
}
}
}
static void Main(string[] args)
{
// change the no of people crossing the bridge
// add or remove corresponding time to the list
int n = 4;
List<int> band = new List<int>() { 1, 2, 5, 10 };
band.Sort();
Console.WriteLine("The total time taken to cross the bridge is: " + Program.TotalTime(band, n));
Console.ReadLine();
}
}
OUTPUT:
The total time taken to cross the bridge is: 17
For,
int n = 5;
List<int> band = new List<int>() { 1, 2, 5, 10, 12 };
OUTPUT:
The total time taken to cross the bridge is: 25
For,
int n = 4;
List<int> band = new List<int>() { 5, 10, 20, 25 };
OUTPUT
The total time taken to cross the bridge is: 60

Answer 5

Here's the response in ruby:
#values = [1, 2, 5, 10]
# #values = [1, 2, 5, 10, 20, 25, 30, 35, 40]
#values.sort!
#position = #values.map { |v| :first }
#total = 0
def send_people(first, second)
first_time = #values[first]
second_time = #values[second]
#position[first] = :second
#position[second] = :second
p "crossing #{first_time} and #{second_time}"
first_time > second_time ? first_time : second_time
end
def send_lowest
value = nil
#values.each_with_index do |v, i|
if #position[i] == :second
value = v
#position[i] = :first
break
end
end
p "return #{value}"
return value
end
def highest_two
first = nil
second = nil
first_arr = #position - [:second]
if (first_arr.length % 2) == 0
#values.each_with_index do |v, i|
if #position[i] == :first
first = i unless first
second = i if !second && i != first
end
break if first && second
end
else
#values.reverse.each_with_index do |v, i|
real_index = #values.length - i - 1
if #position[real_index] == :first
first = real_index unless first
second = real_index if !second && real_index != first
end
break if first && second
end
end
return first, second
end
#we first send the first two
#total += send_people(0, 1)
#then we get the lowest one from there
#total += send_lowest
#we loop through the rest with highest 2 always being sent
while #position.include?(:first)
first, second = highest_two
#total += send_people(first, second)
#total += send_lowest if #position.include?(:first)
end
p "Total time: #{#total}"

Answer 6

Another Ruby implementation inspired by #roc-khalil 's solution
#values = [1,2,5,10]
# #values = [1,2,5,10,20,25]
#left = #values.sort
#right = []
#total_time = 0
def trace(moving)
puts moving
puts "State: #{#left} #{#right}"
puts "Time: #{#total_time}"
puts "-------------------------"
end
# move right the fastest two
def move_fastest_right!
fastest_two = #left.shift(2)
#right = #right + fastest_two
#right = #right.sort
#total_time += fastest_two.max
trace "Moving right: #{fastest_two}"
end
# move left the fastest runner
def move_fastest_left!
fastest_one = #right.shift
#left << fastest_one
#left.sort!
#total_time += fastest_one
trace "Moving left: #{fastest_one}"
end
# move right the slowest two
def move_slowest_right!
slowest_two = #left.pop(2)
#right = #right + slowest_two
#right = #right.sort
#total_time += slowest_two.max
trace "Moving right: #{slowest_two}"
end
def iterate!
move_fastest_right!
return if #left.length == 0
move_fastest_left!
move_slowest_right!
return if #left.length == 0
move_fastest_left!
end
puts "State: #{#left} #{#right}"
puts "-------------------------"
while #left.length > 0
iterate!
end
Output:
State: [1, 2, 5, 10] []
-------------------------
Moving right: [1, 2]
State: [5, 10] [1, 2]
Time: 2
-------------------------
Moving left: 1
State: [1, 5, 10] [2]
Time: 3
-------------------------
Moving right: [5, 10]
State: [1] [2, 5, 10]
Time: 13
-------------------------
Moving left: 2
State: [1, 2] [5, 10]
Time: 15
-------------------------
Moving right: [1, 2]
State: [] [1, 2, 5, 10]
Time: 17
-------------------------

Answer 7

An exhaustive search of all possibilities is simple with such a small problem space. Breadth or depth first would work. It is a simple CS problem.
I prefer the missionary and cannibal problems myself

Answer 8

17 -- a very common question
-> 1-2 = 2
<- 2 = 2
-> 5,10 = 10 (none of them has to return)
<- 1 = 1
-> 1,2 = 2
all on the other side
total = 2+2+10+1+2 = 17
usually people get it as 19 in the first try

Answer 9

Considering there will be 2 sides, side 1 and side 2, and N number of people should cross from side 1 to side 2. The logic to cross the bridge by a limit of L number of people would be -
Step 1 : Move L number of the fastest members from side 1 to side 2
Step 2 : Bring back the fastest person back from Side 2 to Side 1
Step 3 : Move L number of slowest members from side 1 to side 2
Step 4 : Bring back the fastest person among the ones present in Side 2
Repeat these steps until you will be left with no one in Side 1, either at the end of step 2 or at the end of step 4.
A code in C# for n number of people, with just 2 persons at a time is here. This will intake N number of people, which can be specified in runtime. It will then accept person name and time taken, for N people. The output also specifies the iteration of the lowest time possible.
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace RiverCrossing_Problem
{
class Program
{
static void Main(string[] args)
{
Dictionary<string, int> Side1 = new Dictionary<string, int>();
Dictionary<string, int> Side2 = new Dictionary<string, int>();
Console.WriteLine("Enter number of persons");
int n = Convert.ToInt32(Console.ReadLine());
Console.WriteLine("Enter the name and time taken by each");
for(int a =0; a<n; a++)
{
string tempname = Console.ReadLine();
int temptime = Convert.ToInt32(Console.ReadLine());
Side1.Add(tempname, temptime);
}
Console.WriteLine("Shortest time and logic:");
int totaltime = 0;
int i = 1;
do
{
KeyValuePair<string, int> low1, low2, high1, high2;
if (i % 2 == 1)
{
LowestTwo(Side1, out low1, out low2);
Console.WriteLine("{0} and {1} goes from side 1 to side 2, time taken = {2}", low1.Key, low2.Key, low2.Value);
Side1.Remove(low2.Key);
Side1.Remove(low1.Key);
Side2.Add(low2.Key, low2.Value);
Side2.Add(low1.Key, low1.Value);
totaltime += low2.Value;
low1 = LowestOne(Side2);
Console.WriteLine("{0} comes back to side 1, time taken = {1}", low1.Key, low1.Value);
totaltime += low1.Value;
Side1.Add(low1.Key, low1.Value);
Side2.Remove(low1.Key);
i++;
}
else
{
HighestTwo(Side1, out high1, out high2);
Console.WriteLine("{0} and {1} goes from side 1 to side 2, time taken = {2}", high1.Key, high2.Key, high1.Value);
Side1.Remove(high1.Key);
Side1.Remove(high2.Key);
Side2.Add(high1.Key, high1.Value);
Side2.Add(high2.Key, high2.Value);
totaltime += high1.Value;
low1 = LowestOne(Side2);
Console.WriteLine("{0} comes back to side 1, time taken = {1}", low1.Key, low1.Value);
Side2.Remove(low1.Key);
Side1.Add(low1.Key, low1.Value);
totaltime += low1.Value;
i++;
}
} while (Side1.Count > 2);
KeyValuePair<string, int> low3, low4;
LowestTwo(Side1, out low3, out low4);
Console.WriteLine("{0} and {1} goes from side 1 to side 2, time taken = {2}", low3.Key, low4.Key, low4.Value);
Side2.Add(low4.Key, low4.Value);
Side2.Add(low3.Key, low3.Value);
totaltime += low4.Value;
Console.WriteLine("\n");
Console.WriteLine("Total Time taken = {0}", totaltime);
}
public static void LowestTwo(Dictionary<string, int> a, out KeyValuePair<string, int> low1, out KeyValuePair<string, int> low2)
{
Dictionary<string, int> b = a;
low1 = b.OrderBy(kvp => kvp.Value).First();
b.Remove(low1.Key);
low2 = b.OrderBy(kvp => kvp.Value).First();
}
public static void HighestTwo(Dictionary<string,int> a, out KeyValuePair<string,int> high1, out KeyValuePair<string,int> high2)
{
Dictionary<string, int> b = a;
high1 = b.OrderByDescending(k => k.Value).First();
b.Remove(high1.Key);
high2 = b.OrderByDescending(k => k.Value).First();
}
public static KeyValuePair<string, int> LowestOne(Dictionary<string,int> a)
{
Dictionary<string, int> b = a;
return b.OrderBy(k => k.Value).First();
}
}
}
Sample output for a random input provided which is 7 in this case, and 2 persons to cross at a time will be:
Enter number of persons
7
Enter the name and time taken by each
A
2
B
5
C
3
D
7
E
9
F
4
G
6
Shortest time and logic:
A and C goes from side 1 to side 2, time taken = 3
A comes back to side 1, time taken = 2
E and D goes from side 1 to side 2, time taken = 9
C comes back to side 1, time taken = 3
A and C goes from side 1 to side 2, time taken = 3
A comes back to side 1, time taken = 2
G and B goes from side 1 to side 2, time taken = 6
C comes back to side 1, time taken = 3
A and C goes from side 1 to side 2, time taken = 3
A comes back to side 1, time taken = 2
A and F goes from side 1 to side 2, time taken = 4
Total Time taken = 40

Answer 10

I mapped out the possible solutions algebraically and came out the with the fastest time . and assigning algebra with the list of A,B,C,D where A is the smallest and D is the biggest
the formula for the shortest time is B+A+D+B+B or 3B+A+D
or in wordy terms, the sum of second fastest times 3 and add with the Most Fastest and Most Slowest.
looking at the program there was also a question of increased items. Although I haven't gone through it, but I am guessing the formula still applies, just add till all items with the second item times 3 and sum of everything except 2nd slowest times.
e.g. since 4 items are 3 x second + first and fourth.
then 5 items are 3 x second + first, third and fifth.
would like to check this out using the program.
also i just looked at the pdf shared above, so for more items it is the sum of
3 x second + fastest + sum of slowest of each subsequent pair.
looking at the steps for the optimized solution, the idea is
-right - for two items going to the right the fastest is 1st and 2nd fastest ,
-left - then plus the fastest going back for a single item is the fastest item
-right - bring the slowest 2 items, which will account for only the slowest item and disregard the second slowest.
-left - the 2nd fastest item.
-final right - the 1st and 2nd fastest again
so again summing up = 2nd fastest goes 3 times, fastest goes once, and slowest goes with 2nd slowest.

Answer 11

A simple algorithm is : assume 'N' is the number of people who can cross at same time and one person has to cross back bearing the torch
When moving people from first side to second side preference should be given to the 'N' slowest walkers
Always use fastest walker to take torch from second side to first side
When moving people from first side to second side, take into consideration who will bring back the torch in the next step. If the speed of the torch bearer in next step will be equal to the fastest walker, among the 'N' slowest walkers, in the current step then instead of choosing 'N' slowest walker, as given in '1', choose 'N' fastest walkers
Here is a sample python script which does this: https://github.com/meowbowgrr/puzzles/blob/master/bridgentorch.py