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If at any point during this strategy one person encounters the wall, the other person stops whatever they were doing and instead walks perpendicularly directly towards the wall.

Let the starting point have coordinates of $(0,0)$. In Phase 1 of the strategy, one person walks in a straight line to $(frac{1}{sqrt{3}},-1)$, and the other walks to $(-frac{1}{sqrt{3}},-1)$.

The people then proceed straight to $(frac{sqrt{3}}{2},-frac{1}{2})$ and $(-frac{sqrt{3}}{2},-frac{1}{2})$ (they will not find the wall while doing this).

In Phase 2 each person walks in an arc of $frac{3}{2}sin^{-1}frac{3}{4}-frac{pi}{3}$ radians (approximately 12.886 degrees)* around the circle, and then continues on a tangent until reaching the line $x+y=sqrt{2}$ or $-x+y=sqrt{2}$.

Then, the people walk along the lines $x+y=sqrt{2}$ and $-x+y=sqrt{2}$ until they reach $(frac{sqrt{2}}{2},frac{sqrt{2}}{2})$ and $(-frac{sqrt{2}}{2},frac{sqrt{2}}{2})$ (again, they will not find the wall until they reach these points). In Phase 3, they walk along the circle towards $(0,1)$, which they will reach at a time of $sqrt3+frac{8sqrt2-2sqrt7}{5}+frac{3}{2}sin^{-1}frac{3}{4}-frac{pi}{12}approx3.94679$ hours if the wall is not encountered earlier.

The worst case if the wall is found during Phase 1 is $1+frac{2}{sqrt3}approx2.15470$ hours.

The worst case if the wall is found during Phase 2 is when the wall is at an angle of $frac{1}{2}sin^{-1}frac{3}{4}$ radians (approximately 24.295 degrees)* from the vertical. The farther person reaches it at a time of $1+sqrt3+frac{sqrt7}{2}+frac{3}{2}sin^{-1}frac{3}{4}-frac{pi}{3}approx4.27892$ hours.

The worst case if the wall is found during Phase 3 is $1+sqrt3+frac{8sqrt2-2sqrt7}{5}+frac{3}{2}sin^{-1}frac{3}{4}-frac{pi}{3}approx4.16139$ hours (if the wall is found immediately after they reach the circle, the farther person takes $1$ hour to reach it).

Overall, the worst possible case takes $boxed{4.27892}$ hours.

Diagrams: green circle shows where the wall is found, green diamond is where the other person reaches it.

Worst case for phase 2 (and worst case overall):

Worst case for phase 3:

Latest possible wall encounter:

*Where does this strange value $sin^{-1}frac{3}{4}$ come from?

Name the starting point $A$ and the unit circle $O$. Suppose that the people are following paths that are mirror images across the x-axis. Call the horizontal tangent ($y=-1$) $l$. Suppose the wall (call it $w$ is at an angle of $theta$ to the vertical. Consider which choice of path minimizes the time so that:

• Both people touch $l$.


• Then, one person touches $w$ without going inside $O$ (if they go inside the circle, they are missing some tangents).


• Then, the other person stops following the mirror-image path and heads to $w$.

Note that if we let $v$ be the reflection of $w$ across the x-axis, the time for the last step (second person moves to $w$) is the same as if the first person moved to $v$ instead.

Applying the reflective technique I used in this answer, we now reflect $A$ over $l$ to get $A^prime$, $v$ over $w$ to get $v^prime$, and then $A^prime$ and $O$ over $v^prime$ to get $A^{primeprime}$ and $O^prime$:

Now what is half the length of the shortest path that passes from $A^prime$ to $A^{primeprime}$ while staying outside $O$ and $O^prime$?

If $frac{pi}{9}<theta<frac{pi}{6}$, the path looks like this:

$ABA^prime$ is a right triangle with $AB=1$ and $AA^prime=2$, so $angle A^prime AB=frac{pi}{3}$ and $A^prime B=sqrt3$. $AC$ is parallel to $v^prime$, so $angle A^prime AC=3theta$ and $angle BAC=3theta-frac{pi}{3}$. But what is $CD$?

Reflect $A$ over $w$ to get $E$, and construct rectangle $CDGF$ with $E$ on $FG$. $angle AEG=pi-2theta$ because $EG$ is perpendicular to $v^prime$ and $AE$ is perpendicular to $w$. $AE=2$, and $EG=1$. So $CD=FG=EG+EF=1+2cos{2theta}$.

In total, the path length from $A^prime$ to $D$ is $sqrt3+3theta-frac{pi}{3}+1+2cos{2theta}$. When we take the derivative and set it to zero, we get $3-4sin{2theta}=0$, and so the required path length is maximized at $theta=frac{1}{2}sin^{-1}frac{3}{4}$ with a value of $sqrt3+frac{3}{2}sin^{-1}frac{3}{4}-frac{pi}{3}+1+frac{sqrt7}{2}approx4.27982$, as desired.

A slight improvement on Electric_monk's answer:

each one starts off 90 degrees from eachtother for $sqrt2$ miles straight. Then they walk for 1 mile straight in such a direction so they are exactly 2 miles apart and on opposite sides of the starting point, then they follow the circle of radius 1 mile until they meet each other. Like this:

I believe the worse case is here when they are the 2 miles apart and one of them finds it, which means that the answer is $1+sqrt2 +2 approx 4.41$

4.57 hours.

They walk 1 mile in the same direction, then one walks clockwise and the other walks counter-clockwise along an arc of 1 mile radius.
Worst case: They started parallel to the wall. In which case, one of them would reach the wall when he completes a quarter circle and then the other would have to walk 2 miles to the first guy.

The other answer is shorter, but relies upon both being able to hold a very precise arc while blind. If they can do that, great, however walking in a straight line may be easier.

In which case the answer is approximately 6 hours. Assuming they're able to maintain orientation on a straight course, and can accurately turn to a desired heading, and can accurately track time.

They can walk the complete bounding box of a 1 mile radius circle in that time:

1. For the first hour, they should both walk together in the same direction.


2. Then one turns 90 degrees to the right and the other 90 degrees to the left.


3. Both walk straight for the next hour.


4. Each makes another 90 degree turn in the same direction (left turns left again, right turns right again).


5. Both walk straight for the next two hours.


6. Repeat step #4.


7. Both walk straight for the next hour, and which point they are reunited.


8. If at any point after step #1 one of them encounters the wall, they can transmit to the other that they've found it; the one receiving the message will 1) execute a 180 degree turn if messaged during hour 2, or 2) execute a 90 degree turn if messaged during hours 3 or 4, or 3) continue on their current heading otherwise.

In the worst-case scenario, the wall will be encountered at one of the corners reached at the end of the 4th hour. The other participant will need to walk 2 hours to reach that spot.

However, I say "approximately" 6 hours because the bounding box is larger than the 1 mile radius within which some portion of the wall should coincide (at a tangent, I assume, because otherwise some portion of the wall is less than 1 mile away). So if the wall is encountered at a corner of the bounding box, there's a reasonable chance that the second participant will encounter a different portion of the wall somewhat before they reach the far corner (i.e. in slightly less than 6 hours). Both will have reached the wall, but they will be at different points along it.

It will take them 6 hours, 8 minutes, and 30 seconds (Pi+3 hours). I'm assuming that they are capable of traveling in a perfect arc.

Both travelers walk in opposite directions away from each other for one hour. In the best case scenario, one would reach the wall at this point and radio the other. Otherwise, both would head clockwise around the circle. In the worst case scenario, the wall was immediately to the left of one of the wise guys and the other one has to walk half the radius of the circle (taking Pi hours) to reach the wall. They then radio the other, who is on the other side of the circle, to traverse the diameter back.

1 hour to reach the circle, Pi hours to traverse half the circle, 2 hours for the one who didn't reach it to cut back across; Pi+3 hours.

I think both of them can reach the wall in less than 4 hours.

For starters, they begin to walk in the opposite directions. Let us name them as $c$ for closer one and $f$ for further.

Case 1

After 1 hour, if $c$ finds the wall, then $f$ turns back 180 degrees and walks 2 hour. 1 + 2 = 3 hours.

Case 2

If neither finds the wall after 1 hour, then both of them again turns 90 degrees in the opposite directions and walk 1 more hour.

Case 2.1

If the $c$ heads in the right direction, then he will find the wall less than 1 hour. $f$ will compute the angle of his direction, correct his direction and walk through the wall, which will take less than 2 hours.

Case 2.2

If $c$ again heads in the wrong direction, he will not find wall after 1 hour. However, now $f$ will be closer to the wall. $f$ turns another 90 degrees and keeps on walking. $c$ waits $f$ to reach the wall, and then himself walks through the wall.