Fair yut stick throws

Four sticks, each a fair coin

Every throw is four separate sticks, and each one lands flat side up exactly half the time. The result is the number of flat sides showing, just like real sticks. Real wooden sticks are rarely perfectly balanced; in Yunnori every stick is an exact 50/50.

The odds in the table below follow directly from that: Gae is the most common result and Yut and Mo are the rarest.

Real sticks vs Yunnori’s sticks

Real yut sticks are flat on one side and rounded on the other, so they don’t land like a coin. Statisticians have looked at this. A 1996 paper in The Korean Journal of Applied Statistics estimated the odds by experiment, throwing several kinds of store-bought sticks. Statistician Heo Myung-hoi modelled how the half-round stick tips as it lands and put the ratio of flat side up to round side up at about 6 to 4.

Take that 6:4 ratio and assume each stick lands flat side up 60% of the time, independently of the other three. That simple model gives a binomial distribution, and the results shift: Gae and Geol become equally common, Yut comes up about twice as often as in Yunnori, and Mo drops to about one throw in forty. Real sticks thrown together can knock into each other, so their actual odds can differ from this model. The list compares the two (Do includes Back-Do):

• Do: 25% in Yunnori, 15.36% with the 6:4 model
• Gae: 37.5% in Yunnori, 34.56% with the 6:4 model
• Geol: 25% in Yunnori, 34.56% with the 6:4 model
• Yut: 6.25% in Yunnori, 12.96% with the 6:4 model
• Mo: 6.25% in Yunnori, 2.56% with the 6:4 model

Why Yunnori uses even sticks

Real odds change from one set of sticks to the next and with how the sticks are thrown, so there is no single “true” set of odds to copy. Yunnori uses a model that is simple to state and that automated tests check: four even sticks, which give exactly the odds in the table on this page.

Sources

• Park Jin-kyung and Park Heung-sun, “On estimation of the probability of Yut” (윷의 확률 추정에 대하여), The Korean Journal of Applied Statistics 9(2), 1996.
• Kim Hyung-ja, “명절 윷판 속에 이런 비밀이?”, Sisa Journal, 5 February 2013. Reports Heo Myung-hoi’s (Korea University) 6:4 ratio of flat side up to round side up.

Nothing else goes into a throw

The throw doesn’t know who is throwing, what they own, or who is winning. You and the computer throw with the same sticks, and purchases are cosmetic only, so no one can buy better throws.

Offline games: seeded and repeatable

Games against the computer and pass-and-play games draw their throws from a seeded generator in the rules engine. The same seed always produces the same sequence of throws, so a recorded game can be replayed and checked against its seed. The engine includes a verifier that does exactly that, and automated tests check that large samples of throws match the odds on this page.

The one scripted throw is the first throw inside the in-app tutorial, which is fixed to a forward result so the lesson can show a move.

Online games: thrown on the server

In online rooms the phones never pick the throw. The game server throws the four sticks with a cryptographically secure random number generator, and it checks every move against the same rules engine before anyone sees it.

The six throws: 도·개·걸·윷·모·빽도

Odds assume four sticks that each land flat side up half the time, with Back-Do on (the standard setting). With Back-Do off, the marked-stick result counts as Do, so Do becomes 4/16.

Frequently asked questions