يحل صديقي اللغز بدون تلميح.

Because Arabic very often uses verb–subject–object order, especially in Modern Standard Arabic.

So:

• يحل صديقي اللغز = literally solves my friend the puzzle
• Natural English order is My friend solves the puzzle

You could also say:

• صديقي يحل اللغز

That is also correct, but it gives the subject صديقي a bit more prominence. Both are good Arabic.

يحل is the present/imperfect tense verb, here meaning he solves.

A few useful points:

• The basic dictionary form is often given as حَلَّ = to solve, to untie, to unravel
• In this sentence, يحل means he solves
• The يـ at the beginning often marks he in the present tense

So:

• يحل = he solves / he is solving

In context, Arabic present tense can sometimes translate as:

• solves
• is solving
• sometimes even will solve, if the context points to the future

Because the verb tells you the subject is he, but it does not tell you who that he is.

So:

• يحل = he solves
• يحل صديقي = my friend solves

Arabic verbs often contain subject information, but you still use a noun like صديقي when you want to name the subject explicitly.

It is made of:

• صديق = friend
• ـي = my

So:

• صديق = friend
• صديقي = my friend

The ending ـي is a attached possessive pronoun meaning my.

This is very common in Arabic:

• كتابي = my book
• بيتي = my house
• أخي = my brother

Also, once a noun has a possessive ending like ـي, it is already definite, so you do not add الـ.
So الصديقي would be wrong here.

الـ is the Arabic definite article, meaning the.

So:

• لغز = a puzzle / a riddle
• اللغز = the puzzle / the riddle

In this sentence, the noun is definite, so the meaning is the puzzle.

It is also the direct object of the verb يحل, because it is the thing being solved.

In full grammatical analysis, اللغز is in the accusative case because it is the direct object.

Fully vocalized, it would be:

• اللغزَ

The final ـَ (fatḥa) marks the accusative.

But in normal Arabic writing, short vowels are usually not written, so you just see:

• اللغز

This is extremely normal. Native texts usually leave those case endings unwritten.

بدون means without.

It introduces a phrase that tells you something is done in the absence of something else:

• بدون تلميح = without a hint

After بدون, the following noun is treated like it comes after a preposition, so in full vocalization it would be in the genitive:

• بدون تلميحٍ

But again, those short vowel endings are usually omitted in everyday writing.

Because the meaning is general: without a hint or without any hint.

So:

• تلميح = a hint
• التلميح = the hint

In this sentence, the indefinite noun sounds natural because the speaker is not referring to one specific known hint, but to hints in general.

That is why بدون تلميح means:

• without a hint
• without any hint

No. تلميح is a good formal word meaning hint or allusion, but Arabic can use other words depending on context.

For example:

• بدون تلميح = without a hint
• من دون مساعدة = without help
• من دون أي إشارة = without any clue/sign
• من دون تلميحات = without hints

So تلميح is correct, but other choices are possible depending on exactly what nuance you want.

Yes. The root ح-ل-ل has several related meanings in Arabic, and context decides which one is intended.

Depending on context, حلّ can relate to ideas like:

• solving a problem
• undoing or untying something
• loosening
• resolving

In this sentence, because the object is اللغز (the puzzle / the riddle), the meaning is clearly solves.

So the context makes the intended meaning obvious.

A careful fully vocalized pronunciation would be:

يَحُلُّ صَدِيقِي اللُّغْزَ بِدُونِ تَلْمِيحٍ

A few notes:

• يَحُلُّ = yaḥullu
• صَدِيقِي = ṣadīqī
• اللُّغْزَ = al-lughza
• بِدُونِ = bidūni
• تَلْمِيحٍ = talmīḥin

In ordinary unvoweled writing, those short vowels are simply not shown:

يحل صديقي اللغز بدون تلميح

Yes. The Arabic present/imperfect tense often covers both:

• simple present: my friend solves
• present progressive: my friend is solving

Which English version sounds best depends on context.

So this sentence could mean:

• My friend solves the puzzle without a hint
• My friend is solving the puzzle without a hint

If the surrounding context is about a current action happening right now, the second translation may be more natural in English.