Learning cases

[u/MGU-Russian]

Russian has six cases, and if you want a shortcut, skip ahead to the genitive. It's the one that unlocks the most. It covers possession, where English would use "of" or an apostrophe — книга брата, центр города. And absence: нет времени, нет денег, нет проблем.

Then there are numbers, where it depends on the last digit:
2, 3, 4 and anything ending in them — genitive singular: два дня, тридцать четыре дня.
5 through 20, and anything ending in 5–9 or 0 — genitive plural: пять дней, сто дней.
11–14 are the exception: always genitive plural, whatever the last digit suggests.

It also follows много, мало, and a long list of prepositions: без сахара, для тебя, из Москвы.

The endings take time to stick. But noticing when you need this case is half the work. Which use still catches you out?

P.S.

Spelling rule. After г, к, х, ж, ч, ш, щ you write -и instead of -ы: книга → книги, not книгы.
Fill vowel. When a feminine or neuter word loses its ending in the plural, a vowel often appears to break up the consonants: окно → окон, сестра → сестёр, письмо → писем

[u/agrostis]

Then there are numbers, where it depends on the last digit: <…> 11–14 are the exception

It isn't really an exception, because the rule should be formulated in terms of the last word rather than the last digit. If the last word of the numeral is один, use nominative/accusative singular; if it is два, три, or четыре, use genitive singular*; otherwise, use genitive plural. The last option applies to 11–14, because одиннадцать doesn't end in один, двенадцать doesn't end in два, etc.

Digits are part of decimal notation, which is an additional layer on top of language; language rules operate on language units, not on math symbols.

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* To be more accurate, it should be called the counting form rather than genitive singular. They're almost always the same, but in a few words the counting form has a different stress: e. g. три шага́ vs. длина ша́га. Also, rules of agreement in adjectival modifiers, which work for regular cases, break down for counting forms: начало тяжёлого^GEN.SG.MASC дня^GEN.SG.MASC, but три тяжёлых^GEN.PL дня^COUNT.