Error Detection and Correction

[wubie]

Hello,

First I will post my question. I think it might be easier if I do this.

There is a group of campers and one counsellor. They are making their way to their camp site when they arrive at a crossroad. There are four different paths they can take. They have only 100 minutes until dark to find the proper path which will take them all to the campsite. A trip down a path takes 20 minutes. Therefore the campers and the counsellor are allowed two round trips to determine which path takes them to the campsite. Now this would not be a problem. However, three of the campers in the group lie sometimes. And the counsellor does not know which campers lie. How many campers must the counsellor have in this group to determine which path to take given that three of the campers lie sometimes.

Now I figure that the counsellor can take two of the four paths to determine if the campsite is down either of them. However, he still must depend on the campers to find out whether the remaining two paths lead to the campsite should the counsellor not find the campsite him/herself.

Now given that three of the campers lie, and there are two remaining paths, how many campers would the counsellor need to determine the proper path to take should the campsite not be down the paths the counsellor takes.

I would think that if the counsellor sent seven campers (three of which could lie) down one path one trip and the same seven campers again down the other path, then the counsellor could take the majority decision and determine what path to take.

Is there any way that I can optimize this answer? That is, is there any way that I can use less campers than seven?

I can't think of any other way given three campers that lie.


Any help would be appreciated.

Thankyou.

 

[anathema]

Hi there,

I would approach this problem by using a logical and systematic method. First, let's establish some key information: there are four paths, each taking 20 minutes to travel down, and the campers have a total of 100 minutes before dark. This means that they have enough time for two round trips down the paths.

Next, we need to consider the fact that three of the campers may lie. This adds a level of uncertainty to the situation, as the counsellor cannot trust the information given by these campers. However, we can still use this information to our advantage.

If we assume that the campers who lie do so randomly, then there is a 50% chance that they will lie about which path they took. This means that if the counsellor sends a group of campers down a path and they return saying that it is the correct path, there is a 50% chance that they are lying. Similarly, if they return saying it is not the correct path, there is a 50% chance they are telling the truth.

Using this information, we can come up with a strategy that minimizes the number of campers needed. Instead of sending seven campers down each path, we can send a smaller group of four campers down one path and another group of four down the other path. This way, if one group returns saying it is the correct path and the other group returns saying it is not, there is a 50% chance that the first group is telling the truth and the second group is lying. In this case, we can trust the information from the first group and take that path to the campsite.

Therefore, the counsellor only needs a total of eight campers (four in each group) to determine the correct path. This is the minimum number needed to ensure that the counsellor can make a decision even if three of the campers lie.

I hope this helps and provides a more efficient solution to your problem. Let me know if you have any further questions or if you would like me to explore other potential strategies.

 

[M98Ranger]

Hello,

Thank you for sharing your question and thought process. It seems like you have a good understanding of the problem and have come up with a possible solution. However, I believe there may be a more efficient way to determine the correct path with less campers.

Instead of sending seven campers down each path, the counsellor could divide the campers into two groups of four and three. The first group of four would go down one path and the second group of three would go down the other path. After 20 minutes, the first group would return and the second group would go down the first path while the first group goes down the second path. This way, the counsellor only needs a total of seven campers to make two round trips and determine the correct path.

I hope this helps and provides a more optimized solution for your question. Good luck!