Semel talum et ter aleam iacio.

Because they are direct objects of iacio.

The verb iacio means I throw, so the thing being thrown must be in the accusative case:

• talus → talum
• alea → aleam

So the sentence uses the forms that mean a talus / knucklebone and an alea / die as the objects of the verb.

Iacio is the 1st person singular present active indicative form of iacere.

So it means:

• I throw
• or, depending on context, I am throwing

In this sentence, it is the main verb for both parts:

• semel talum iacio
• ter aleam iacio

Latin often leaves the second iacio unstated because it is easy to understand from the first one.

They are adverbs.

• semel = once
• ter = three times

They modify the action of throwing, not the nouns. In other words, they tell you how many times the action happens.

So:

• semel talum iacio = I throw the knucklebone once
• ter aleam iacio = I throw the die three times

Because they are adverbs, they do not change form to match talum or aleam.

It is really joining two parallel phrases:

• semel talum
• ter aleam

Both depend on the same verb, iacio.

So the structure is essentially:

• semel talum iacio et ter aleam iacio

But Latin does not need to repeat the verb the second time. This is very common.

Because Latin word order is much more flexible than English word order.

A very common Latin pattern is to put the verb last, especially in straightforward statements. So:

• Semel talum et ter aleam iacio

sounds natural in Latin.

English usually needs a stricter order, but Latin can move words around more freely because the endings show their grammatical role.

Yes. Since the case endings make the grammar clear, Latin can often rearrange the words without changing the basic meaning.

For example, forms like these would still be understandable:

• Iacio semel talum et ter aleam
• Talum semel et aleam ter iacio

The exact word order can affect emphasis or style, but not the core meaning.

The version you have is a normal Latin way to present the sentence.

Here aleam is singular because the sentence is talking about one die being thrown three times, not several dice at once.

So:

• aleam = one die
• aleas = several dice

English can be a little confusing here because dice is often used very broadly, but Latin is being precise with the singular.

Yes. Talus can have more than one meaning.

It can refer to:

• the ankle bone
• a knucklebone
• a gaming piece made from such a bone

In this sentence, because it is something being thrown, the gaming sense is the relevant one.

This is normal in Latin: one word can have several related meanings, and the context tells you which one is intended.

Because Latin often has special adverbs for this idea.

Some very common ones are:

• semel = once
• bis = twice
• ter = three times
• quater = four times

So in Latin, using these single-word adverbs is the normal and idiomatic way to express repetition.

Yes, but only in the sense that Latin is omitting a repeated verb that is already understood.

A fuller version would be:

• Semel talum iacio et ter aleam iacio

But Latin regularly avoids repeating words unnecessarily. English does this too sometimes, but Latin does it very comfortably.

So the sentence is perfectly complete as it stands.