"Breed new monsters! Select individuals with different elements. Complex combinations are unstable and may turn out like a parent."
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The Breeding Structure is a nearly mandatory structure on every island. It allows you to the combine the elements of two parents (inputs) into one single child (output).
A quick reference chart for breeding new monsters can be found here.
General Breeding Rules
Monsters of the same species cannot breed. If a monster is placed on one side of the Breeding Structure, all other monsters of the same type will be greyed out, and unable to be selected on the other side.
When breeding two eligible parents, their elements are combined into a pool. If there are no elements repeated, then the offspring could potentially be a new species, consisting of all of the elements combined. If a repetition occurs, the resulting offspring will usually be one of the parents. Even with a correct combination, it is still possible to have an offspring that matches one of the parents.
Example: Breeding a Maw (Water, Cold) with a Clamble (Plant, Earth, Cold) contributes 4 different elements to the pool (Plant, Earth, Cold, Water).
Although the Entbrat consists of Plant, Earth, Cold, and Water elements, it cannot be bred with this combination because the Cold element was repeated.
Since there were repeated elements, the resulting offspring will be one of the parents: most likely the Maw (2 elements) because it has fewer elements.
This is because elements are repeated, and the 4-element monsters cannot be the result of "incorrect" breeding combinations. The 4-element monsters can only be the result of valid breeding combinations.
The only times you can successfully breed monsters that have elements in common are when you're trying to get one of the special monsters such as the Ethereal Monsters, the Shugabush, or the Seasonal Monsters. And these breeding attempts still usually fail, giving an offspring that matches one of the parents.
Double Element Monsters
Generally the easiest to breed and hatch, two-element monsters simply require two single-element monsters. For example, a Tweedle and a Potbelly should breed to create a Dandidoo. These breeding combinations appear to have 100% success rate.
It should be noted that breeding double element Ethereal Monsters does not have a 100% success rate.
Triple Element Monsters
These are slightly harder to breed and hatch, as a two-element monster and a single-element monster are required, and sometimes the offspring will take the species of a parent instead. For example, if the breeding is successful, a Quibble and Mammott would create a Congle.
Quad Element Monsters
The hardest of all natural monsters to breed and hatch, most of the time you will breed a parent by accident.
The player can breed these by pairing 2 two-element monsters or breeding a three-element and a single-element monster together. Here are two examples of trying to make a Deedge:
The combinations of three-element and one-element monsters have higher success rates for breeding a Quad Element Monster than using a pair of two-element monsters. Otherwise, there currently is no evidence that any specific combination for any monster is more successful than another.
However when breeding for Quad Element monsters, many will choose which combination based on the breeding times of the parents, so that "failures" can be moved from the Breeding Structure to the Nursery at predictable or regular intervals without using diamonds to speed the process.
Using a pair of two-element monsters to breed a Deedge: Pango and Oaktopus
With each failed attempt, regardless of which parent is returned, the result is a monster that breeds in 8 hrs, so every 8 hrs an egg will hatch from the Nursery and one can be moved from the Breeding Structure to the Nursery.
With each failed attempt, the result will be an egg that takes 12 hrs or one that takes 2 minutes. This can result in a Mammott egg sitting complete in the Breeding Structure waiting nearly 12 hrs for a Spunge egg to finish its time in the Nursery. Since the result of a failed attempt is random, it can be hard to predict how often eggs will be ready to be moved - if several Mammott eggs are returned in a row, then several breeding attempts can happen in a matter of minutes. However if a Spunge egg is returned, then no more attempts can be made for 12 hrs (unless diamonds are used).
As such, the method used depends on the player's goal. If the goal is to breed a Quad Element monster, 3/1 combinations are recommended due to a higher success rate. If the goal is to sell unwanted monsters for money, then breeding a pair of two-element monsters can result in more regular intervals while still providing the chance of breeding a Quad Element Monster.
If a Quad Element Monster is put in the Breeding Structure with another monster, the result of the breeding attempt will ALWAYS be that other monster (the only exception has been when certain Limited Time Monsters are available, or when trying to breed Ethereal Monsters). Thus if breeding to sell unwanted monsters, you can breed a target monster with the Quad Element Monster and know exactly what the results will be. This can be used to sell monster eggs at regular intervals if the same combination (or one with equivalent breeding times) are used repeatedly. For example: Bowgart and Entbrat will always produce Bowgarts, except a very small chance of producing the Ghazt.
Limited Time Monsters
Some monsters do not follow regular breeding requirements, instead requiring specific monsters as parents and a combination that is usually considered "unstable". Examples include the Punkleton, a limited Halloween monster that was bred using a Bowgart and a T-Rox, and the Yool, which is bred during the Christmas season with Thumpies and a Congle.
Limited Time Monsters are harder to breed than non-limited monsters, leading some to presume that the combinations are not correct.
Similar to Seasonal Monsters, the Legendary Monsters or "Shugafam" require specific breeding combinations. A comprehensive list of these combinations can be found here. The chance of a successful breeding is not 100%, and does not appear to depend on the element level of the non-Shugabush monster.
Wubbox and the Ethereals
With the addition of the Wubbox and then the Ethereal Monsters, it appears there may be fundamental, and permanent, changes to the breeding rules. The Wubbox added another element, but as of yet cannot be placed in the breeding structure. The Ethereal monsters appear to be the result of breeding an island's 4-element monster with ANY of the 3-element monsters available (although given the rarity of the monsters, not every combination has been confirmed). This obviously results not only in duplicated elements (like the Limited Time monsters do), but also up to four different end combinations (unlike the Limited Time Monsters which only have one combination).
The Ethereals appear to be even rarer than even Limited Time monsters, with suggestions that the chance of success is around 1%.
A Rare Monster and the common counterpart may be bred together. This may be useful especially if you are trying to obtain rare Rare Monsters such as Rare Entbrat, although one Rare Entbrat would need to be bred first.
Bonus Breeding Structure
Spent cash and got this. The following is about a premium structure. Please only edit this if you are eligible of obtaining it on your island(s).
"The Bonus Breeding Structure provides a vital advantage when special Monsters with limited availability are up for grabs! The Monster-Handlers would like to extend a heartfelt thank-you for your support of My Singing Monsters! They couldn't do it without you!"
Beginning with Version 1.3.3, it's possible for users who have previously purchased at least one item for the game to buy a second Breeding Structure for each Island, at a cost of 40 diamonds. It's also possible to buy a Bonus Nursery to pair with it.
Enhanced Breeding Structure
"Breeding is tricky business at the best of times, so every little advantage helps. Thanks to the invaluable field research that has been collected over the past two years, the complex interplay of elemental powers is now better understood than ever before! This Enhanced Breeding Structure cuts breeding wait times by 25%!"
With version 1.3.0, it became possible to upgrade the Breeding structure by spending 25 gems, and doing so reduces breeding wait times by 25%.
The Enhanced Breeding Structure is the same size and shape as the normal one, but the big pink heart becomes golden, and there are other golden touches added to the structure. This upgrade is available in all mobile-device versions of the game, and does not require any real-money purchase.
Since the breeding and hatching times for any given monster are the same, and an egg must go through both processes one after the other (and there's no way to get rid of an unwanted egg without hatching it), there is no benefit to enhancing the Breeding Structure unless you also enhance the Nursery.