Turtle Racing 競亀

Turtle racing (競亀 Kyousou Kame) is a gambling game.

It is played in the Gambling Den in Gion (賭場 Toba).
Kiryuu has to find or buy a turtle and then submit it in to a race.
Kiryuu can then bet on which turtle(s) he thinks will win the race. It doesn't have to be his own turtle!

Original instructions using ThePatrick's full Kenzan FAQ/walkthrough on gamefaqs

Completion

Race them turtles. Win them races. Win all 63 races. Each turtle has a rank. There are a set amount of races in each rank (10 or 13). You have to win every race in every rank to complete this achievement. The ranks are as follows:

Rank (Kanji)	Rank (Romaji)	Rank	Races won per Rank
平幕	Hiramaku	5th rank	00/10
小結	Komusubi	4th rank	00/10
関脇	Sekiwake	3rd rank	00/10
大関	Ozeki	2nd rank	00/10
横綱	Yokozuna	Champion	00/10
無差別	Musabetsu	Indiscriminate	00/13

The turtle racing competition follows the sumo wrestling rules.
Each turtle starts at the 5th rank. When it wins 3 races, it goes up a rank.
To check: (?? What happens if it subsequently loses races ?? does it's rank go down?? If not then that means you will need at least 4 turtles to get to champion rank??)
You can enter your turtle into 9 races of the Indiscriminate rank at any time. The other 4 Indiscriminate races are unlocked by completing special conditions detailed in the race section below.

Turtle Racing and Keeping

Turtle Keeping

To keep turtles you're going to need a bucket. Get one from the Turtle vendor. He's located on Shijou Street in the Rakugaichou suburbs. Halfway from the Gion Gate to Hanami Street is a man with a handkercheif on his head standing behind a limbo pole. This is the Turtle Vendor. He sells turtles and turtle food.

The first time you talk to him there will be an option to get a turtle bucket for free. Now when you go back to the Tatsuya you will have a bucket which will be placed right next to your Save Point.

Once you have a turtle bucket you can now keep them:

You use your turtles in the turtle racing in the Gambling Den.

You can give the turtles food, to raise their statistics.

The turtles will run away if you don't take care of them (give them food).

They have a set life span and will die.

Each turtle has several abilities. Some of their abilities can be altered by feeding them and racing.

You can get an estimate of their usefulness by paying the old guy at the bar in Gion (Sakaya Nakanokura).

Collecting turtles

Picking up turtles you previously flipped. These are available to pick up from Chapter 5 onwards, though you flip a few before. Look in the locations where you flipped a turtle and you should see a shiny glimmer like a locker key. Pick it up and you'll have a turtle in your Current Items.

Buy them off the Turtle Vendor on Shijou Street

By completing the entire Yuujo hostess part of the game. You'll receive an Eternal Turtle. That turtle will take longer to die than normal turtles (it will die), but it won't run away. Its stats are not very good though.

Turtle flipping

You will find turtles on their back in the following places:

Chapter	Name	Name	Where
One	ミヤモトガメ	Miyamotogame	At night, round the back side of the hut that Marume is in.
Two	テラガメ	Teragame	Middle of the temple garden when hiding with Majima.
Three	タガメ	Tagame	In the rice field outside Ukiyo's house on the east bank.
Seven	カワラガメ	Kawaragame	North side of the Riverbank