Ein roter Bauklotz liegt unter dem Tisch.

Because Bauklotz is the subject of the sentence, so it is in the nominative case.

In masculine nominative singular, after ein- words like ein, the adjective takes -er:

• ein roter Bauklotz

This is part of German adjective endings. A useful comparison:

• ein roter Bauklotz = nominative masculine
• ich sehe einen roten Bauklotz = accusative masculine

So roter is correct here because the block is the thing doing the “lying.”

German often prefers a position verb where English might simply use is.

• liegen = to lie, to be lying
• stehen = to stand
• sitzen = to sit

So:

• Der Bauklotz liegt unter dem Tisch.

sounds more natural in German than using ist.

You can say ist unter dem Tisch in some contexts, but liegt is more specific and idiomatic when an object is resting horizontally or simply located somewhere.

Because this sentence describes location, not movement.

The preposition unter is a two-way preposition, which means:

• dative for location: unter dem Tisch = under the table
• accusative for movement toward a place: unter den Tisch = to under the table / underneath the table

Compare:

• Ein roter Bauklotz liegt unter dem Tisch.
  The block is already there.

• Ich lege den Bauklotz unter den Tisch.
  I put the block under the table.

Because Tisch is masculine: der Tisch.

After unter in a location meaning, German uses the dative case, and the dative singular form of der is dem:

• nominative: der Tisch
• dative: dem Tisch

So:

• unter dem Tisch

is the correct form.

You usually learn German nouns together with their article:

• der Bauklotz
• der Tisch

Unfortunately, grammatical gender in German often has to be memorized. Sometimes there are patterns, but many nouns just need to be learned individually.

A good habit is to never learn a noun alone:

• not just Bauklotz
• but der Bauklotz

That makes it much easier to form sentences correctly later.

In a normal German main clause, the finite verb goes in position 2.

Here the sentence begins with the subject:

• Ein roter Bauklotz = position 1
• liegt = position 2
• unter dem Tisch = the rest

So the structure is:

• Subject + verb + rest

This is one of the most important German word-order rules.

Yes. German is flexible about what comes first, as long as the finite verb stays in second position.

So you can also say:

• Unter dem Tisch liegt ein roter Bauklotz.

That version puts more emphasis on the location.

Notice that liegt still stays in second position.

Because all nouns are capitalized in German, not just at the beginning of a sentence.

So:

• ein roter Bauklotz
• der Tisch

This is a standard spelling rule in German.

Bauklotz means a building block, often a wooden toy block.

Its plural is:

• der Bauklotz
• die Bauklötze

Notice the vowel change:

• o → ö

Example:

• Zwei rote Bauklötze liegen unter dem Tisch.

A plural version would be:

• Rote Bauklötze liegen unter dem Tisch.

Changes:

• ein disappears, because there is no indefinite article in the plural
• roter becomes rote
• Bauklotz becomes Bauklötze
• liegt becomes liegen

So:

• singular: Ein roter Bauklotz liegt unter dem Tisch.
• plural: Rote Bauklötze liegen unter dem Tisch.

Because after dem in the dative masculine, the adjective takes -en.

For example:

• unter dem roten Tisch

Here, dem already clearly marks the noun as masculine dative, so the adjective uses the weaker ending -en.

Compare:

• ein roter Bauklotz
• unter dem roten Tisch

This difference is a normal part of German adjective declension.

It comes from the verb liegen.

Present tense:

• ich liege
• du liegst
• er/sie/es liegt
• wir liegen
• ihr liegt
• sie/Sie liegen

Since Ein roter Bauklotz is singular, the sentence uses:

• liegt

No. Unter is a two-way preposition, so it can take either:

• dative for location
• accusative for movement toward a place

Examples:

• Der Ball ist unter dem Stuhl.
• Ich schiebe den Ball unter den Stuhl.

So in your sentence, dem is used because the block is simply located there, not moving there.