The Campers Problem $($Parts $1-3)$

A counselor has $3+2$

campers with her at a junction in a hiking trail $($where $$

is a natural number$).$ She knows their camp is twenty minutes down one of four possible paths. It will be dark in one hour and the group must find their camp before dark. $$

of the $3+2$

campers sometimes lie, and unfortunately the counselor doesn't know which $$

they are. The counselor checks path $4$

$,$ leaving paths $1$

$,2$

$,$ and $3$

for the $3+2$

campers. She sends $+1$

campers down each of paths $1$

and $2$

and sends $$

campers down path $3$

$.$ If she doesn't find the camp down path $4$

she designs a code containing three codewords to deduce the location of the camp. Each codeword will represent the camp being down a particular path either path $1,2,$ or $3.$ Each bit $(0$ or $1)$ in the codeword will represent the answer given by a camper, a $0$

will represent $"$no$$ and a $1$

will represent "yes$.$ Here is the counselor's code:

Part $1$: Find the minimum Hamming distance of the counselors code. Recall that the minimum Hamming distance is the smallest Hamming distance that can be found between any pair of the counselor's codewords. Report your answer in terms of $$

$.$

minH$(,)=$

The feedback should read "The variables found in your answer were: $[$n$]"$

Part $2$: Is the Hamming distance large enough to correct $$

corrupted bits and therefore deduce the location of the camp?

$($No answer given$)$

Part $3$: The counselor figures out that path $4$ is not the correct path. She has an even number of campers $($so $3+2$

is even and therefore $$

is even$)$ with her and needs to figure out if the camp is down path $1$

$,2$

$,$ or $3$

$.$ When she gets back to the junction she asks everyone what was found down their path; first asking everyone that went down path $1$

$,$ then path $2$

$,$ and finally path $3$

$.$ The responses were recorded in three groups with bits and listed in order $($the first group has $+1$

campers, the second has $+1$

campers, and the third has $$

campers$)$:

$10101010101010101000000000$

Where is the camp?