Puzzle 032: A Royal Cleaning

ナゾ０３２　王家の掃除術
友人がある王様の部屋を掃除することに
なったのだが、その部屋には変わった
決まりがある。まず、床のタイルに数字
が書かれていて、１から順に縦が横に
１つずつ移動しながら、全て掃除しな
ければならない。また、数字が書かれた
タイルは、必ずその順番の時に掃除し、
同じタイルは２度と遠てはいけない。
どんな道順で掃除すればいいのだろう？

Google Translate
But as a king to have a friend clean a room, the room has a different rule. First, they have written a number of floor tiles, starting from one vertical to horizontal while moving one by one, all must be cleaned. The tile numbers are written, it must be cleaned at the time of the order, the same tiles should not be two degrees distant. How do I clean it with any directions?

Notes on Puzzle
You're given a 6x6 grid and need to traverse through the grid in a specific order. There are a couple of tiles with numbers of them. You must move through the grid so that when you hit the numbered tiles, that is how many tiles away you are from the start (ie you have to hit the tile numbered 6 as the 6th move). You can only move vertically/horizontally.
When you solve the puzzle, you may notice the path is not as random as it first seems.