That's not true. "I am going to write you" is an old way of saying "I am going to write TO you." It is actually grammatically correct to say either one. Few people nowadays use the old version; it's a little archaic, but it's proper grammar. The "to" is implied, which means that it is understood to be there even if you don't explicitly say it.
If you won't take my word for it, you can check out this site: <http://www.answers.com/Q/Do_you_say_I_write_you_or_i_write_to_you>
True, it is awkward, but nevertheless okay to say.
Running for to come out of the rain is also old English but no one says it anymore. As far as I'm aware, "I will write you" is an american translation of the Spanish sentence "te voy a escribir" into English words, like a lot of americanisms. The website you have posted a link to is a US site, one of the categories is MATH not MATHS
This is just a shortened version of "I am not going to write you a letter". (I will use "a letter" here and not "a postcard" simply because I'm used to the phrase "send you a postcard" in this context.)
Because the direct object is implied, the sentence may seem awkward.
"I am not going to write to you" also leaves out the direct object.
I am not going to write you (implied direct object) -> I am not going to write you a letter (direct object specified)
I am not going to write to you (implied direct object) -> I am not going to write a letter to you (direct object specified)
That is correct. It is American usage; as far as I know it has nothing to do with its being archaic or otherwise. But it could be something the early settlers took over to America. Does anyone know? I have never known it to be used in spoken or written English in the UK. I have been marking students' essays ( in the UK ) all my life and never encountered it. I am not being pedantic, just interested.
MargretheAnton: Yes, there are rules about the placement of object pronouns - plenty of them! I'll try to sort them out for you.
1. When there is only one verb in the sentence, the object pronoun must be placed before the conjugated verb: "Te veo" (I see you (informal)).
1a. If the sentence is negative, the object pronoun follows the "no": "No te veo" (I don't see you).
1b. If you have an indirect and a direct pronoun, the indirect comes first: "Te lo traeré" (I will bring it to you).
1c. You cannot have a combination of two pronouns that start with "L." So, if you wanted to say something like: "I gave it to her" (WRONG: La lo di) you have to change the indirect pronoun to "se" (CORRECT: Se lo di) and you might want to add something like "a ella" to clarify the sentence (Se lo di a ella).
1d. If you add a reflexive pronoun to the mix, the order becomes:
...reflexive pronoun + indirect pronoun + direct pronoun... (Phew!) Think RID.
(Sorry. Can't think of an example right now!)
2. If there is a conjugated verb and an infinitive, you can attach the pronoun(s) to the end of the infinitive or leave them before the conjugated verb. Examples: "Quiero verte" or "Te quiero ver" (I want to see you); "No quiero verte" or "No te quiero ver" (I don't want to see you); "Te lo traeré" or "Traerételo" (I will bring it to you. Note the accent on the "é" to maintain pronunciation).
3 You can also attach the pronoun(s) to the end of a command or a gerund. See more here: http://www.dummies.com/how-to/content/placing-spanish-object-pronouns-correctly.html