Kings and Legends Combine Optimization

Often when combining cards, I want to know how many cards are most efficient for combining.

Sometimes you might be close to the last tier and you want to use all your cards to get 80-100% chance because you want to almost guarantee success.  Other times when combining lower cards you just want to minimize losses.  Is it more wasteful to use 4 cards for a 60% chance or 5 cards for a 80% chance?  There is no clear choice especially when the probabilities for combining one tier of cards is different than probabilities at higher tiers.

Let’s consider 5 attempts:

4 cards with 60% success rate: 5 attempts x 4 cards consumed == 20 consumed.  Out of 5 attempts 60% succeed, so that gives an average of 3 cards.  So it cost us 20 cards to make 3 cards of a higher tier.  That’s 20/3 == 6.66 cards per higher tier card.

5 cards with 80% success rate: 5 x 5 == 25 consumed.  Out of 5 attempts 80% success, so we get 4 cards.  25/4==6.25

So using 5 cards will give us an average of 6.25 cards consumed per higher tier card.  This is a lower cost than the 6.66 when using 4 cards per combine, so using 5 is more efficient.

To simplify this calculation, you can just take the number of cards and divide by the success rate(as a decimal).  So 5 at 80% == 5/ 0.8 == 6.25.  And we see 4 at 60% == 4/0.6 == 6.66. If combining 6 cards gives 100%, then 6/1.0 == 6 which is a lower cost than the 6.25.  So don’t forget to do the calculation for the 100% case as well, as it might be the most efficient.

Conclusion:

So in conclusion it’s easy to calculate the average cost where N is the number of cards being combined, P is the success % as a decimal, and C is the average cost per successful combined.

N / P = C

Just add one card at a time, and perform this calculation for each probability, and whichever cost is lower is the best choice.

Note: This logic doesn’t apply to blacksmith enchanting, because failure doesn’t just cost you the gems you used in the attempt, it can also cost you the previous upgrade since failure can result in a downgrade.  This makes it more challenging to optimize.  It seems better to aim for much higher success rate on the enchants to avoid a downgrade, except the first one since there is no chance of downgrade.