Nadie pudo adivinar el número.

Because nadie is already a negative word meaning nobody / no one.

In Spanish, if the negative word comes before the verb, you do not add no:

• Nadie pudo adivinar el número. = Nobody could guess the number.

But if the negative word comes after the verb, then you normally do need no:

• No pudo adivinar el número nadie.

So the rule is:

• Negative word before the verb → no extra no
• Negative word after the verb → use no

Pudo is the preterite form of poder for él / ella / usted.

Since nadie is grammatically treated as third-person singular, the verb also goes in third-person singular:

• nadie pudo
• not nadie pudieron

The preterite pudo suggests a completed past situation: at that moment or in that situation, nobody managed to guess it.

This is a very common question, because both can translate as could in English.

• pudo = managed to / was able to in a specific completed situation
• podía = could / used to be able to / was able to in a more ongoing, descriptive, or habitual sense

So:

• Nadie pudo adivinar el número.
  = Nobody was able to guess the number / Nobody managed to guess the number.

This sounds like a specific event with a clear result.

If you said:

• Nadie podía adivinar el número.

it would sound more like Nobody could guess the number in the sense of an ongoing situation, perhaps because it was too difficult.

Because after poder, Spanish normally uses an infinitive.

Structure:

• poder + infinitive

Examples:

• Puedo hablar. = I can speak.
• No pudieron entrar. = They couldn’t get in.
• Nadie pudo adivinar el número. = Nobody could guess the number.

So pudo adivinar literally works like was able to guess.

Adivinar usually means to guess.

Depending on context, it can also mean things like:

• to figure out
• to predict
• to guess correctly

In this sentence, the most natural meaning is simply:

• to guess the number

For example:

• Adivina mi edad. = Guess my age.
• No pude adivinar la respuesta. = I couldn’t guess the answer.

El número means the number, referring to a specific number that both speaker and listener understand from context.

• el número = the number
• un número = a number

So this sentence is talking about one particular number, not just any number.

Compare:

• Nadie pudo adivinar el número. = Nobody could guess the number.
• Nadie pudo adivinar un número. = Nobody could guess a number.
  This sounds less natural in most contexts unless you really mean any number at all.

Because número has the stress on the first syllable: NÚ-me-ro.

In Spanish spelling rules, a word ending in a vowel, n, or s is normally stressed on the second-to-last syllable unless there is a written accent. Without the accent, numero would be pronounced differently.

So the accent mark shows the correct stress:

• número

This accent is part of the correct spelling.

It is grammatically singular, even though it refers to zero people.

That is why the verb is singular:

• Nadie pudo...
• not Nadie pudieron...

This is similar to English nobody or no one, which also usually take singular agreement.

Yes, but the most neutral and common order here is:

• Nadie pudo adivinar el número.

You can change the structure for emphasis, but then other parts of the sentence may change too. For example:

• El número no lo pudo adivinar nadie.
  = Nobody could guess the number.
  This emphasizes the number.

Notice that when nadie comes after the verb, you need no, and in this version you also often use lo to refer back to el número.

So yes, word order can change, but the original sentence is the simplest and most straightforward version.

Yes. In this sentence, both are natural:

• Nobody could guess the number.
• No one could guess the number.

In English, the difference is mostly stylistic. No one can sound slightly more formal or neutral, while nobody is very common in everyday speech.

Because el número is a thing, not a person.

The personal a is generally used before a specific human direct object:

• Veo a María. = I see María.

But with things, you usually do not use it:

• Adivinar el número
• Veo la casa

So el número does not take the personal a here.