Das Turnier ist eine Begegnung dreier Schulen.
auf allen vieren kriechen
Can someone please explain to me the grammar behind using numbers as adjectives? Examples would help a lot.
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Adjectives can be used in three ways:
Die lauten Tenöre verlassen die Bühne.
The loud tenors leave the stage.
Die Tenöre sind laut.
The tenors are loud.
Die Tenöre singen laut.
The tenors sing loudly.
But also some other word can be used in these functions, for example numerals perfectly qualify as attributes:
Die drei Tenöre verlassen die Bühne.
The three tenors leave the stage.
Here the nominal group »die drei Tenöre« was the subject, i.e. in nominative case. But you can also use this group as an attribute of another noun. To do so, you just need to transfer »die drei Tenöre« into genitive case and attach it as an attribute at the right side of another noun (here: Auftritt):
Der Auftritt der drei Tenöre begeisterte das Publikum.
The performance of the three tenors thrilled the audience.
But you also can talk not about "the" three tenors, but about any three tenors. In this case you don't have an article, and the genitive case is marked by a modification of the attribute:
Der Auftritt dreier Tenöre begeisterte das Publikum.
The performance of three tenors thrilled the audience.
So, in such a construction the word »dreier« is still a numeral, not an adjective. But although it's not an adjective it still is used as an attribute of a noun (»Tenöre«). And with this noun it builds a nominal group (»drei Tenöre«) which itself is the genitive attribute of another noun (»Auftritt«), and »dreier« is just the genitive form of the numeral »drei«.
In you other example is an error. This is the correct version:
auf allen Vieren kriechen
crawl on all fours
Die Vier is a noun (in English: the fours). So it's neither a numeral nor an adjective.