On a $100 \times 70$ meter football field, the players have buried the referee's whistle. The referee starts from one corner of the field and can take 1-meter steps in any direction. After each step, they will tell him whether he has gotten closer to the whistle or not. If he gets within 1 meter of the whistle, he can already see it.

What should he do? Also, give an estimate of how many steps it will take for him to find the whistle, if he was originally $d$ meters away from it.
Let the corners of the field be denoted by $A, B, C, D$, with $A$ being the corner from which the game leader starts. If the game leader cannot see the flag from $A$, they should head towards the opposite corner $C$. Continue moving straight along the $A C$ line until the flag is seen or until they are told that the last step moved them further away from the flag. In the latter case, denote the starting point of the last step by $P$ and the endpoint by $Q$. Return from $Q$ to $P$, and from there, take a step perpendicular to the $A C$ line. If this brings them closer to the flag, continue moving straight until they see it. Otherwise, move in the opposite direction along the line perpendicular to $A C$ until they see the flag.

![](https://cdn.mathpix.com/cropped/2024_05_02_2b1d9e03cca6c116cb37g-1.jpg?height=513&width=689&top_left_y=470&top_left_x=707)

If the game leader sees the flag while moving along the $A C$ line, it is clear that they will take at most $[d+1]$ steps. (As usual, $[x]$ denotes the greatest integer not greater than $x$.) Otherwise, let $q$ be the perpendicular bisector of the segment $P Q$, and $p$ be the reflection of $q$ over $P$. Since stepping into $Q$ moved the game leader away from the flag, the flag is on the side of $q$ that contains $P$. Since the last step into $P$ did not move them further away from the flag, the flag is on the side of $p$ that contains $P$, so it must be within the strip bounded by $p$ and $q$. Denote the endpoint of the first step taken perpendicular to $A C$ by $R$. If this brings the game leader closer to the flag, the flag is on the side of the $A C$ line that contains $R$; otherwise, it is on the opposite side. In either case, by moving along the center of the strip, the game leader cannot pass by the flag without seeing it, since the boundaries of the strip are half a meter from the center line, and they can spot the flag from a meter away. If, for example, they step into $S$ last and see the flag from there, then $A S \leq d+1$, and $A P + P S \leq A S \cdot \sqrt{2}$, which means the number of steps required is at most 4 more. (Two steps are needed to turn around from $Q$, and two more if they need to turn around from $R$.) Therefore, the game leader will find the flag in at most $[\sqrt{2}(d+1)] + 4$ steps.

Note. Consider the part of the circle centered at $A$ with radius $d$ that lies within the rectangle $A B C D$. Cut this part in half with the line $A C$, and let $\delta$ be the central angle corresponding to the larger part. If $\delta < 100$, the game leader generally needs to take fewer steps with the above strategy compared to starting from $A$ towards $D$ and then continuing the search along a line perpendicular to $A D$. Specifically, if $d = A C$, our strategy requires 122 steps, while starting along $A D$ requires 170 steps. Denote by $L(T)$ the number of steps required for a given strategy when the game leader finds the flag at $T$, and let $M(d)$ be the maximum of the $L(T)$ values for all $T$ such that $A T = d$. Let $m(d)$ be the minimum of the $M(d)$ values over all possible strategies. We have only determined the value of $m(d)$ for $d = A C$, and determining the entire function $m(d)$ is a very difficult task.

If $d$ is not too small, it is advantageous for the game leader to take a probing step in every direction and then head towards the flag based on the responses to these steps. The problem can also be examined on an infinitely large field. It is a very difficult question to determine whether there exists a number $K$ and a strategy such that for any large $d$, $[d + K]$ steps are sufficient to find the flag.