Question 12
Mila stands on an infinitely large board divided into squares and starts moving. An $n$-jump is a movement in which Mila moves one square left, right, up or down and then $n$ squares in a direction perpendicular to that. Below is an example where Mila starts in the middle box on the left and first does a $1$-jump, followed by a $2$-jump and then a $3$-jump, $4$-jump and $5$-jump.
Suppose Mila first does a $1$-jump, then a $2$-jump, then a $3$-jump, a $4$-jump, and so on. Finally, she does a $m$-jump. For which positive integers $m$ can Mila choose her $m$ jumps such that she can get back to her starting square?
Suppose Mila first does a $1$-jump, then a $2$-jump, then a $3$-jump, a $4$-jump, and so on. Finally, she does a $m$-jump. For which positive integers $m$ can Mila choose her $m$ jumps such that she can get back to her starting square?