Placenta in mensa est, sed mater dicit eam post cenam dandam esse.

Because in can take either the ablative or the accusative, depending on the meaning.

• in + ablative = in/on a place, showing location
• in + accusative = into/onto a place, showing motion toward

Here, mensa is ablative singular, so in mensa est means it is on the table. There is no motion; the cake is simply located there.

Eam means her/it and refers back to placenta. Latin often uses a pronoun instead of repeating the noun, just as English would say it instead of repeating the cake.

It is:

• feminine singular
• accusative

It has to be feminine because placenta is feminine.

Because after dicit in this kind of construction, Latin uses an indirect statement with:

• a subject accusative
• an infinitive

So in mater dicit eam post cenam dandam esse, the clause literally works like:

• mater dicit = mother says
• eam ... esse = it ... to be

In English we say Mother says that it must be given after dinner, but Latin expresses this with accusative + infinitive instead of that + finite verb.

So eam is accusative because it is the subject of the infinitive esse.

Because Latin regularly uses the accusative-and-infinitive construction after verbs of saying, thinking, knowing, hearing, and so on.

So instead of saying something shaped like:

• Mother says that it should be given

Latin often says:

• Mother says it to be given / to have to be given

That is why you get dicit ... esse.

A direct statement might be:

• Placenta post cenam danda est = The cake must be given after dinner

But when that becomes indirect after dicit, Latin changes it to:

• mater dicit eam post cenam dandam esse

Dandam esse is a form of the passive periphrastic, which expresses necessity, obligation, or something that is to be done.

It is made from:

• a gerundive: dandam
• a form of esse: esse

The verb behind it is do, dare = to give.

So:

• danda est = must be given, is to be given
• dandam esse = to have to be given, to be needing to be given

In this sentence, it means that the cake should be given/served after dinner.

Because the gerundive must agree with the noun or pronoun it describes.

Here it agrees with eam, which refers to placenta:

• feminine
• singular
• accusative

So the gerundive is also:

• feminine
• singular
• accusative

Hence dandam.

If the noun were masculine, neuter, plural, or in another case, the gerundive form would change to match.

Post cenam means after dinner.

The preposition post takes the accusative case, so cena becomes cenam.

This is just a normal prepositional phrase:

• post + accusative = after

So:

• post cenam = after dinner
• literally after the dinner

By itself, a gerundive often carries the idea of needing to be done or being to be done. When combined with esse, it usually expresses necessity fairly clearly.

So dandam esse is stronger than just to be given in a neutral sense. It usually means something like:

• to have to be given
• must be given
• should be given
• is to be given

The exact English wording depends on context. In this sentence, should be served/given after dinner sounds very natural.

Latin does not have to state every detail if it is understood from context.

So eam post cenam dandam esse simply means that it is to be given after dinner. The sentence does not specify the recipient.

If Latin wanted to say to the children, for example, it could add a dative:

• eam pueris post cenam dandam esse

But here the focus is just on when the cake is to be given.

Yes. Although the basic verb dare means to give, English often prefers a more natural verb depending on context.

Since this is food, a smoother translation may be:

• Mother says it should be served after dinner

That is not a different Latin meaning so much as a more idiomatic English rendering.

The word order is normal for Latin. Latin word order is more flexible than English because the endings show the grammatical relationships.

The sentence is:

• Placenta in mensa est
• sed mater dicit eam post cenam dandam esse

A few things to notice:

• Placenta comes first, which gives it prominence.
• sed mater introduces the contrast: the cake is on the table, but mother says...
• esse comes at the end, which is very common in Latin subordinate infinitive clauses.
• dandam is placed close to esse, since together they form the passive periphrastic idea.

So the order may feel different from English, but it is not strange in Latin.

It breaks down like this:

• Placenta in mensa est = The cake is on the table
• sed = but
• mater dicit = mother says
• eam post cenam dandam esse = that it must/should be given after dinner

So the grammar is:

1. a main statement
2. a contrast with sed
3. a verb of saying
4. an indirect statement using accusative + infinitive

This is a very common Latin pattern, so it is a useful sentence to study carefully.