Luck titles guide

Overview
The Unlucky and Lucky titles are account-based titles which are obtained through various in-game activities:


 * Playing the Shing Jea Boardwalk games Nine Rings and Rings of Fortune. These games last for certain special events and will return in the future.
 * Opening chests with a lockpick. Retaining the lockpick adds points to the Lucky title; breaking it adds points to the Unlucky title. This is a permanent feature of the game.
 * Using a Four-Leaf Clover and having 15% DP removed from the character awards 4 Lucky points. The clovers were part of the Lucky weekend event.
 * Participating and winning in the the ring game during the finale of the Wintersday 2006 event awarded Lucky points.

Detailed analyses for the two main activities, the Boardwalk games and treasure hunting, follow:

Boardwalk games analysis

 * Nine Rings
 * Corners win 55 tickets, while center only win 25 tickets and the edges 40 tickets.
 * The amount of tickets gained is statistically equal in every location, however more games are lost on corners (awarding more Unlucky points and the same amount of Lucky points), so the following analysis will be based on playing on a corner ring.
 * Rings of Fortune
 * All locations are equally advantageous.

Achieving one title only

 * If one wants to achieve the Unlucky title with the least amount of Gold spent, Rings of Fortune is optimal.
 * Otherwise, Nine Rings is better. Notably, it is faster for either title.

Achieving both titles at the same time

 * It is beneficial to play both games in order to reach both titles of the same tier at the same time.

Lockpick analysis
A Lockpick has a certain chance of breaking after opening a chest. If a lockpick breaks, it will add 25 points to the Unlucky title; otherwise it will add 250 points to the Lucky title. These numbers are the same in Normal and Hard Mode.

The rate at which (Un)Lucky points are gained depends on the chance of retaining lockpicks and the number of chests opened per hour. The latter depends on the difficulty of the area, the amount of chests there and the skill of the player/team. It cannot be quantified and will be omitted in this analysis.

The following formulas are relevant to the lockpick analysis:


 * Let R be the chance to retain a lockpick after use. R is the character level divided by 2, plus 3 times the character's treasure hunter title rank, plus 2 times the character's lucky title rank. for a level 20 character R is at least 10% and at most 41% (unconfirmed and assuming the max lucky title rank is 5).


 * The average number of lucky points per chest is 250 * R (The amount of lucky points per retained lockpick times the chance of retaining the lockpick)
 * The average number of unlucky points per chest is 25 * (1 - R) (The number of unlucky points per broken lockpick times the chance of breaking the lockpick)


 * The price per chest is 1500*(1 - R)

As for the price of gold per (Un)Lucky point, we can calculate:


 * ''Price per Lucky point = 1500 (1 - R) / (250 * R) = 6 * (1 - R) / R (Average price per chest divided by average number of lucky points per chest)
 * ''Price per Unlucky point = 1500 (1 - R) / (25 * (1 - R) = 60 (Average price per chest divided by average number of unlucky points per chest)

These results mean the following:
 * The price of Unlucky points is constant for all values of R, because as R increases, the price of opening a chest and the unlucky points per chest drop at the same rate.
 * The price of Lucky points decreases quadratically as R increases, because the price per chest drops and at the same time the number of Lucky points per chest increase.
 * Example:
 * At the minimum R = 0.1, 1000 Lucky points cost 54,000 on average.
 * At R=0.5, 1000 points cost 6,000 (it is unconfirmed whether this R can be reached).
 * At R=0.98, 1000 Lucky points cost only 122 (it is unconfirmed whether this R can be reached).


 * Therefore, the higher the chance to retain a lockpick, the better.
 * For illustration, these are the most extreme prices to reach the maximum Lucky title, starting from zero Lucky points (the calculations include improving retention rates from increasing Lucky and Treasure hunter title rank):
 * At Treasure hunter rank 7 and only picking Ascalonian chests on Normal mode, 10.600 chests have to be opened and 887 have to be spent.
 * At Treasure hunter rank 0 and only picking Hard mode chests, 26.300 chests have to be opened and 24.450 have to be spent.