Konstantinov N.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_6741038a32a996d2ce33g-61.jpg?height=838&width=786&top_left_y=1353&top_left_x=639)

It is easy to see that similarly, observers can be arranged and the snail's movement can be set such that in $t$ minutes ($t>1$), it will crawl

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

Let's prove that this is the maximum displacement of the snail.

Let $a_{1}$ be the first observer. Consider all observers who started watching the snail either when $a_{1}$ finished or even earlier (such observers exist by the problem's condition). 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 them as $a_{3}$. 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 observing earlier, then there are observers who started watching later than $a_{k}$ began, so we can choose 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 $a_{1}, a_{3}$, $a_{5}, \ldots$ do not overlap; similarly, the intervals in which observers $a_{2}, a_{4}$, $a_{6}, \ldots$ watched do not overlap.

Indeed, if, for example, there were a moment in time when both $a_{1}$ and $a_{3}$ were observing, this would mean that observer $a_{2}$ 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

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

Since the selected set of observers covers the entire observation interval, the snail could not have crawled more than the sum of the displacements recorded by all 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 that 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_6741038a32a996d2ce33g-62.jpg?height=489&width=786&top_left_y=807&top_left_x=636)

Fig. 2

## Answer

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

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