Panis secandus est; deinde frusta inter puellas dividenda sunt.

This is the passive periphrastic: a gerundive plus a form of esse.

• secandus est = must be cut / is to be cut
• dividenda sunt = must be divided / are to be divided

The key idea is necessity or obligation. Latin often uses this construction where English would use must, should, or needs to be.

So:

• Panis secandus est = The bread must be cut
• frusta ... dividenda sunt = the pieces must be divided ...

The gerundive is a verbal adjective, so it behaves like an adjective and agrees with a noun.

Because the gerundive agrees with the noun it goes with in:

• gender
• number
• case

So:

• panis is masculine singular nominative
• therefore the gerundive is secandus: masculine singular nominative

And:

• frusta is neuter plural nominative
• therefore the gerundive is dividenda: neuter plural nominative

This is just like an ordinary adjective agreeing with its noun.

So the structure is:

• panis secandus est
• frusta dividenda sunt

Because in this construction the thing that must be done to something becomes the grammatical subject of the sentence.

In English, we say:

• The bread must be cut

Here the bread is also the subject of the English sentence, even though it receives the action.

Latin does the same thing:

• Panis secandus est

So panis is nominative because it is the subject of est.

Because the verb esse agrees with the subject:

• panis = singular, so est
• frusta = plural, so sunt

That gives:

• Panis secandus est
• frusta dividenda sunt

Even though the gerundive expresses the idea of action, the finite verb here is still just esse, and it must agree in number with the subject.

They are not the same.

• secatur means is being cut or is cut
• secandus est means must be cut or is to be cut

So:

• Panis secatur = The bread is being cut
• Panis secandus est = The bread must be cut

The first describes an action happening. The second expresses necessity or obligation.

That difference is very important.

A gerundive is a verbal adjective. That means it comes from a verb, but it behaves grammatically like an adjective.

Here:

• secandus comes from secare = to cut
• dividenda comes from dividere = to divide

Because gerundives are adjectives, they must agree with the nouns they describe.

In this sentence, the gerundive does not simply mean cutting or dividing. In combination with esse, it gives the sense needing to be cut/divided.

So:

• secandus = needing to be cut
• dividenda = needing to be divided

Frusta is the neuter plural form of frustum, meaning piece, chunk, or bit.

So the sentence first says:

• Panis secandus est = The bread must be cut

Then:

• deinde frusta ... dividenda sunt = then the pieces must be divided ...

So the idea is:

1. cut the bread
2. then divide the resulting pieces

Because frusta is neuter plural, the gerundive must also be neuter plural:

• frusta dividenda sunt

Yes, inter often means between, but it can also mean among when more than two people are involved.

So:

• inter puellas = among the girls

Also, inter takes the accusative case, which is why we get puellas, not puellis.

So this phrase is:

• inter
  • accusative
• inter puellas = among the girls

English speakers often expect something like a dative here, but Latin commonly uses inter + accusative for the idea of distribution among people.

Not in the same way.

A dative can sometimes be used with ideas of giving or assigning, but here the sentence is emphasizing distribution among the girls, and inter puellas is a very natural way to express that.

So:

• inter puellas focuses on the idea among the girls
• puellis would suggest more simply to/for the girls

Both ideas may be close in meaning in some contexts, but they are not identical in structure or emphasis.

In the passive periphrastic, the person who has the obligation can be put in the dative. This is often called the dative of agent.

For example:

• Nobis panis secandus est = We must cut the bread
  literally, To us the bread must be cut

And:

• Nobis frusta inter puellas dividenda sunt = We must divide the pieces among the girls

So in your sentence, no agent is stated. It just gives the obligation in an impersonal way: the bread must be cut; the pieces must be divided.

Deinde means then, next, or after that.

It shows sequence:

1. Panis secandus est = first, the bread must be cut
2. deinde frusta inter puellas dividenda sunt = then, the pieces must be divided among the girls

So deinde helps organize the steps of the action.

Yes. Both are possible, depending on context.

The Latin passive periphrastic often has a range like:

• must be cut
• ought to be cut
• should be cut
• is to be cut

In many beginner translations, must be is the clearest way to show the force of the construction, because it makes the idea of necessity very obvious.

So Panis secandus est most straightforwardly means:

• The bread must be cut

But in a different context, is to be cut could also be a good translation.