The case for "article + adjective + noun" combination?

Question

The fact that, concerning the same case as well as the same gender, articles and adjectives does not always follow the same pattern (i.e., don't get the same ending) is a bit hard to understand.

• I know that language is not mathematic and is not always logical. But, is there any point that help to understand this difference between article and adjective, despite the same case and gender?
• Should I just memorize all different sentences, or always think of gender, case, and the difference?

Example:

Ich kann euch das in einem anderen Video erklären.

Why that isn't

Ich kann euch das in einem anderem Video erklären.

or

Ich kann euch das in einen anderen Video erklären.

?

Answer 1

The simple answer is that articles (or more generally, determiners) and adjectives belong to different parts of speech and therefore inflect differently. It will all make at least a little bit of sense if you learn about strong and weak adjective declension.

The endings of the strong adjectives are almost identical with those of determiners; in fact, for expository purposes, I'll leave out the genitive singular so that they are in fact identical.

dieses Bier – helles Bier
mit meinem Gewissen – mit gutem Gewissen
aufgrund der Nachfrage – aufgrund großer Nachfrage

In the above examples, there is either a determiner or an adjective present, never both. The adjectives bear strong endings, which look identical to those of determiners.

Strong inflection also occurs when both a determiner and an adjective are present and the determiner bears no ending.

ein/mein/kein/unser/… neues Bier
welch großes Vergnügen

However, if an inflected determiner and an adjective occur together, the adjective must follow the weak declension. Weak declension is really simple: The ending is either -e or -en; -e occurs in the nominative singular, -en otherwise. (Note that for the singular neuter and feminine, the accusative is identical to the nominative and therefore also has -e.)

dieses helle Bier
mit meinem guten Gewissen
aufgrund der großen Nachfrage

Answer 2

Those declination patterns are the result of constant mangling and selection for both distinctiveness and ease of speech. That's why there seems to be no system.

The basic rule is:

• Once there is a distinct case determiner, the other case determiners are selected that way there is no tongue-twister.

See this article for a longer explanation of the same sort.

Answer 3

Mastering case and gender agreement in fluent speech is an immensely tough nut to crack, so to anyone learning German as L2: Kudos; nicht aufgeben!

Two patterns overlap here (I was tempted to write "conspire against the unsuspecting learner", however language evolution is hardly ever deliberate or purposeful):

Strong vs Weak Adjective Declension

David's answer illustrates that difference very well. Nothing to add!

Washed-out Case Markers

While German articles, nouns, and adjectives have conserved case declension, the clarity of case markings has eroded over time. This led to, for instance, only six semantically distinct definite article words covering sixteen functionally distinct case-number-gender roles (four cases times three singular and one collective plural gender). In that effort they're supported - well, sort of - by far too few semantically distinct noun endings like -r, -s, -m, and -n, so that one way or another the combination of article variant and inflected noun does provide just enough information to deduct case and thereby, sentence function.

As if this wasn't complicated enough, the distinction between weak adjective inflection with, and strong adjective inflection without determiner only adds to the conundrum. This is why your assumption that a series of -m endings should consistently indicate a dative case ('einem anderem', theoretically) is perfectly logical, yet unfortunately not correct.

I am being deliberately judgmental:

For one, German's close cousin Dutch has done away with cases except in the context of a few fixed idioms and formulae. The same is true of other Germanic languages other than Icelandic. Russian on the other hand manages well with six (!) cases, but those are identifiable by much clearer markings and not confused by articles; presumably that is similar in other Slavic languages.

For another, I pick up on quite a few article-adjective-noun case agreement errors in the German online press, where authors may not have the luxury of getting content proofread prior to publication, and these things escape easily (no blame).

As multilingual user experience designer I am a stickler for form supporting function across all human artefacts, and comparatively, the German language simply does not do a particularly 'user friendly' job in that regard. My sympathies!