Konstantinov N.

Several people observed a snail for $t$ minutes. Each person observed it for exactly 1 minute and noticed that during this minute the snail crawled exactly 1 meter. At no time was the snail left unobserved. What is the smallest and the largest distance the snail could have crawled in these $t$ minutes?
In Fig. 1, a graph is shown indicating that a snail can crawl 4 meters in 2.5 minutes.

![](https://cdn.mathpix.com/cropped/2024_05_06_696cdbe703e904f1264dg-53.jpg?height=800&width=786&top_left_y=0&top_left_x=636)

Fig. 1

It is easy to see that by arranging the observers and the snail's movement in a similar manner, the snail can crawl

$2(t-1)$ meters in $t$ minutes ($t>1$) if $t$ is an integer, and $2[t]$ meters if $t$ is not an integer.

We will prove that this is the maximum distance the snail can crawl.

Let $a_{1}$ be the first observer. Consider all observers who started watching the snail either when $a_{1}$ finished or earlier (such observers exist by the problem's conditions). Let $a_{2}$ be the last of these observers. Next, consider all observers who started watching the snail no later than when $a_{2}$ finished, and denote the last of these by $a_{3}$. Similarly, select observer $a_{4}$, and so on. Clearly, we will eventually reach an observer who finished observing exactly at the end of the last minute (if observer $a_{k}$ finished earlier, there would be observers who started watching after $a_{k}$ began, and thus we could select observer $a_{k+1}$).

Let $a_{1}, a_{2}, \ldots, a_{k}$ be all the observers selected in this manner. It is clear that the observation intervals of $a_{1}, a_{3}, a_{5}$, ... do not overlap; similarly, the intervals in which $a_{2}, a_{4}, a_{6}, \ldots$ observed do not overlap. Indeed, if there were a moment when both $a_{1}$ and $a_{3}$ were observing, it would mean that $a_{2}$ was chosen incorrectly, as $a_{3}$ started observing later than $a_{2}$ but before $a_{1}$ finished.

Since the observation intervals of $a_{1}, a_{3}, a_{5}, \ldots$ do not overlap, there are fewer than $t$ such observers in $t$ minutes. Therefore, if $t$ is an integer, there are no more than

$t-1$ of them, and if $t$ is not an integer, no more than $[t]$. The same number limits the number of observers in the "even group": $a_{2}, a_{4}, a_{6}, \ldots$

Since the selected set of observers covers the entire observation interval, the snail could not have crawled more than the sum of the distances observed by all the observers, which is no more than $2(t-1)$ if $t$ is an integer, and $2[t]$ if $t$ is not an integer.

Now, let's estimate the number of "odd" observers from below. If all gaps between adjacent odd observers are less than one, the number of odd observers must be greater than $[t / 2]$. This means the snail could not have crawled a distance less than $[t / 2]+1$. Fig. 2 shows how the snail should move to achieve this minimum.

![](https://cdn.mathpix.com/cropped/2024_05_06_696cdbe703e904f1264dg-53.jpg?height=483&width=778&top_left_y=2331&top_left_x=640)

Fig. 2
$[t / 2]+1$ meters; $2(t-1)$ meters if $t$ is an integer, $2[t]$ meters if $t$ is not an integer.