Learn Hardcore German: Wir möchten bei dem Spiel im Park mitmachen. - We would like to take part in the game in the park.

möchten (literally the subjunctive of mögen) is the polite, softer way to express a wish:
- Wir möchten … = We would like to …
  This sounds friendly, polite, and less pushy.
wollen is more direct:
- Wir wollen … = We want to …
  Depending on context and tone, this can sound stronger, even demanding.

In a neutral or polite context (e.g. talking to a teacher, host, or in public), Wir möchten bei dem Spiel im Park mitmachen is more natural than Wir wollen ….

Why is it wir and not uns at the beginning?

German, like English, changes pronouns depending on their grammatical role:

wir = we → subject (nominative case)
uns = us → object (accusative or dative case)

In this sentence, wir is the subject, the ones doing the action (wanting to join in):

Wir möchten … = We would like …

You would use uns only if we were the object of some action, for example:

Man hat uns eingeladen. – They invited us.

Why does mitmachen go to the end of the sentence?

Mitmachen is a separable verb (mit + machen).

Two things are happening here:

Main clause word order (verb-second rule)
In a normal statement, the finite verb (the part that is conjugated) must be in second position:
- Wir (1st position – subject)
- möchten (2nd position – conjugated verb)
Modal verb + infinitive pattern
Möchten behaves like a modal verb. With modals, the other verb (the infinitive) goes to the end of the clause:
- Wir möchten … mitmachen.

Because mitmachen is used here as a whole infinitive (not split), it stays intact and moves to the very end of the sentence.

Is mitmachen always at the end, or is it sometimes split like mit and machen?

It depends on the structure:

No modal verb, simple present:
- Wir machen bei dem Spiel im Park mit.
  Here, mitmachen is split: machen (the conjugated verb) is in second position, mit goes to the end.
With a modal verb (as in your sentence):
- Wir möchten bei dem Spiel im Park mitmachen.
  Here, möchten is conjugated and in second position, so mitmachen stays together as an infinitive at the end.
Perfect tense:
- Wir haben bei dem Spiel im Park mitgemacht.
  The prefix mit- stays glued to the participle: mitgemacht at the end.

Why do we say bei dem Spiel mitmachen and not just das Spiel machen?

Mitmachen means to take part / to join in (an existing activity), while machen just means to do or to make.

bei dem Spiel mitmachen = to participate in / join in the game
You are joining a game that already exists.
das Spiel machen would literally be to do the game and is not how you naturally say take part in the game in German.
It can sound like you are running or organizing the game, depending on context, but it’s not the standard phrase for taking part.

The usual pattern is:

bei + Dativ after mitmachen:
- bei dem Spiel mitmachen
- bei einem Projekt mitmachen
- bei der Aktion mitmachen

Why is it bei dem Spiel and not bei das Spiel?

This is about case after prepositions.

The preposition bei always takes the dative case.

Masculine/neuter singular article in the accusative: das
Masculine singular article in the dative: dem

Spiel is neuter (das Spiel), and its dative singular is also dem:

Nominative: das Spiel
Accusative: das Spiel
Dative: dem Spiel

Because bei needs the dative, you must say:

bei dem Spiel (not bei das Spiel)

Is bei dem correct, or should it be beim?

Both are grammatically correct:

bei dem Spiel – full form
beim Spiel – contracted form (standard, very common)

In everyday spoken and written German, the contraction beim is more natural and more frequent:

Wir möchten beim Spiel im Park mitmachen.

The full form bei dem can sound slightly more formal or can be used for emphasis (e.g. to stress dem in contrast to something else: nicht bei einem anderen Spiel, sondern bei dem Spiel).

Why is it im Park and not in den Park?

This is about two-way prepositions and whether you express location or movement.

The preposition in can take either accusative or dative:

Accusative (Wohin? – where to?): movement into something
→ in den Park = into the park
Dative (Wo? – where?): static location, in a place
→ in dem Park = in the park → contracted to im Park

In your sentence, you are not moving into the park; the location of the event is in the park. So you describe where the game is, not movement:

im Park = in dem Park (dative for location)

Can I change the order and say Wir möchten im Park bei dem Spiel mitmachen?

Yes, that is grammatically correct.

German allows some flexibility in the order of adverbial phrases such as time, manner, and place. Here, you have two place-related phrases:

bei dem Spiel (at the game)
im Park (in the park)

Both orders are possible:

Wir möchten bei dem Spiel im Park mitmachen.
Wir möchten im Park bei dem Spiel mitmachen.

The difference is mainly in emphasis:

First version tends to stress which game (the one in the park).
Second version can sound like you are first answering where (in the park), then specifying at which event there (at the game).

In everyday conversation, the original version is very natural.

Why do we need bei if mitmachen already means “to join in”?

Mitmachen usually appears with bei + Dativ to say what you are joining:

bei dem Spiel mitmachen
bei der Diskussion mitmachen
bei einem Wettbewerb mitmachen

Think of mitmachen as “to join in / to participate”, and bei + Dativ as “at / in (this activity)”.

English often uses just in:

take part *in the game*

German tends to use mitmachen bei:

bei dem Spiel mitmachen

Without bei, it would sound incomplete:

Wir möchten mitmachen. – We would like to join in (something).
This is okay if the context is obvious.
Wir möchten bei dem Spiel mitmachen. – You clearly say what you want to join.

Is mitmachen closer to “play” or to “participate”?

Mitmachen is closer to “participate / join in” than to “play”.

mitspielen = to play along / to play (as a player in a game)
mitmachen = to take part / to participate in an activity (game, project, event, campaign, etc.)

In your sentence:

bei dem Spiel mitmachen — focus on taking part in the event
bei dem Spiel mitspielen — more specifically: to play as a player in that game

Both can be correct, but they emphasize slightly different things.

How would this sentence look as a yes/no question in German?

You keep the same words but move the conjugated verb to the first position:

Statement: Wir möchten bei dem Spiel im Park mitmachen.
Question: Möchtet ihr bei dem Spiel im Park mitmachen? (to “you” plural)
or: Möchtet ihr beim Spiel im Park mitmachen? (using the contraction)

For a question with wir, you would normally not ask about yourselves, but grammatically it would be:

Möchten wir bei dem Spiel im Park mitmachen?
(e.g. thinking out loud or asking the group whether you want to join).