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The aim of this topic is to discuss whether the level 12 of the Winter Spire (special building from the Winter Event 2018 ) is reachable without spending any diamond. The answer is not that easy. In order to answer this question, one has to investigate the probabilities.

1. Which are the requirements for the Winter Spire level 12 ?

2. How to reach these requirements ?

Matchstick formula

In order to reach the Level 12 of the Winter Spire, Level 1 + 11 Upgrades are needed. Completion of all Quests already provides the Level 1 Winter Spire + 3 Upgrades. The remaining 8 Upgrades require 15 Christmas Trees in order to be completed (See Grand Prize Table ). Therefore, it corresponds to 15 x 20 = 300 candles. Since Star 10 Stars are needed to open a present and to burn the matchstick, the following formula applies :

n_stars / 10 = n_candles / k ; with k as matchstick rate ( 1 < k < 3)

n_stars = 300 x 10 / k ; then Star 1000 < n_stars < Star 3000 ; for example if k = 2 then :

n_stars = 300 x 10 / 2 = Star 1500 Stars required when an average of 2 candles / matchstick are burnt.

Now let's assume that Star 1500 Stars were collected by solving all quests and connecting daily :

n_candles / k = 1500 / 10 <=> n_candles = 150 k <=> k = n_candles / 150

Candles / Matchsticks values to reach a given Winter Spire Level depending on Star Stars available :

Final lvl Upg. Trees Candles

k (Star 1500)

%

k (Star 1600)

%
9 5 9 180 1,20 > 99  1,13 > 99
10 6 11 220 1,47 > 99  1,38 > 99
10 6 12 240 1,60 > 99  1,50 > 99
11 7 13 260 1,73 > 99 1,63 > 99
11 7 14 280 1,87 97 1,75 > 99
12 8 15 300 2,00 52 1,88 98


Considering a 2 Candles / Matchsticks ratio, the Level 12 of the Winter Spire may be reached from Star 1500 Stars with a probability of 52 %. With about Star 1600 Stars (Quests, connexions, bonus from presents), the probability rises to 98 % !

Gaussienne




This odds percentage difference is explained by a Matchsticks rates distribution depending on a Gaussian function centered around an average value of k = 2. The standard deviation is estimated at 0,07 for 150 Matchsticks. Odds indicated in the table were determined by integrating all values above the threshold. Odds calculated with k-values obtained from differents numbers of Stars spent gave the following plot.


Probality to reach the Level 12 Winter Spire in function of the total Stars spent :

WinterSpire12 Stars


Comments : From the Star 1490 Stars given by completing all Quests and a daily connection the chances to reach level 12 are about 44 %. With only additionnal Star 50 Stars, the chances almost double at 80 %. In order to optimise the gain of Stars by opening presents, it is recommended to accumulate at least Star 180 Stars before starting opening several presents till the Star 14 Stars, the shuffle or both are revealed ! Then, collect Star Stars again and repeat. Following this method, in the worst case scenarion (1/18 x 2/18 x 2/18 = 1/1458) Star 10 Stars are given back in 16 rounds (only the shuffle is found, the Star 14 Stars and the "Reveal 2" are missing). It means that 9 series of 16 rounds will cost 9 x 16 x 10 = Star 1440 Stars but will give 9 x 10 = Star 90 Stars back ! Assuming that you connect everyday and complete all the Quests, you will gain at least a total of Star 1580 Stars. With this ammount, there are about 95 % chances to complete the Winter Spire level 12. Raw testing indicates a 20 % Star Stars obtained / Stars spent ratio (to be improved). One could reasonnably expect about Star 200-300 additional Stars by opening present. You can also afford to miss 4-5 days of connection during Christmas to celebrate and enjoy with the family !

Conclusion : Eventually enough Star Stars (> 95% chance) must be collected to complete your beautiful and magnificent Winter Spire for Christmas !

Joyeux Noël à tout le monde ! Buon Natale a tutto il mondo ! Feliz Navidad a todo el mundo ! Frohe Weihnachten für die ganze Welt !