Bakayev E.V.

A grasshopper can jump along a strip of $n$ cells by 8, 9, and 10 cells in either direction. We will call a natural number $n$ jumpable if the grasshopper can, starting from some cell, visit the entire strip, visiting each cell exactly once. Find at least one $n>50$ that is not jumpable.
Suppose a grasshopper has jumped over a strip of 62 cells. We will paint the 8 leftmost cells of the strip white, the next 10 cells black, then 8 cells white again, and so on. In total, there will be 32 white cells and 30 black cells. Since the difference in the number of white and black cells is greater than 1, there must have been a jump between white cells. However, such jumps are impossible.

## Answer

$n=62$.