Jumping leprechaun on a grid

A jumping leprechaun is a special chess piece that lives on an infinite square grid. On the first turn it moves one cell horizontally (left or right) and two cells vertically (up or down). On the $$n$$-th turn it moves $$n$$ cells horizontally and $$n+1$$ cells vertically. Both moves need to happen in a turn. Can the jumping leprechaun come back to its starting location? What is the least number of turns required for that?

• To clarify, may it come back to the start after a half-turn, or does the needed amount of turns need to be an integer?
– oAlt
Aug 28, 2021 at 11:46
• You can't have half turns, they need to be integer. Aug 28, 2021 at 11:48

It can

come back in 4.: ur dl dl ur
This is optimal because the leprechaun alternates black/white squares and obviously cannot do it in 2.

• My dinner interrupted me long enough for you to post this, haha. Congrats. +1. A nice simple and quick puzzle. Aug 28, 2021 at 12:34
• That's it you got it. Perhaps it was too easy. Aug 28, 2021 at 12:37
• Possible Followup questions: 1. Can you repeatedly return to the original point an infinite number of times? 2. Can you jump to any arbitrary point from the origin? (At n=1? or at any n? The latter assumes (1).) 3. If so, can you jump from any arbitrary point to any other arbitrary point regardless of your current n? (If (1) and (2) are true, likely (3) easily follows.)
– stux
Aug 28, 2021 at 20:21
• @stux the answer is yes to all. No matter what n is you can always combine two consecutive moves to move one square diagonally: For example, to move 1 square top and left move right and down first and immediately back. This shows hat you can move to any same parity (black or white) square no matter what n. If you need to switch parity make a single move and then take it from there. Aug 28, 2021 at 20:47
• ah! I didn't realize it would be that simple! Thanks @loopy walt!
– stux
Aug 28, 2021 at 21:20