Learn Hardcore Finnish: Viivain ja kynä ovat kirjahyllyn vieressä sohvapöydällä. - The ruler and the pen are on the coffee table next to the bookcase.

The subject here is viivain ja kynä (ruler and pen) – that’s two items together, so the subject is plural. Finnish verbs agree with the number of the subject, so you need ovat rather than on.

Why are viivain and kynä in the singular if there are two items?

Each individual item is singular: one viivain (ruler) and one kynä (pen). Finnish works like English here:

viivain ja kynä = a ruler and a pen / the ruler and the pen
If you had many of each: viivaimet ja kynät = rulers and pens

So coordination (X and Y) does not automatically make each noun plural; it just makes the whole subject plural, which is why the verb is ovat.

Where are the words the or a in this sentence?

Finnish does not have articles like English a/an or the. The bare noun viivain can mean:

a ruler (introducing it for the first time)
the ruler (if it’s clear from context which one)

The same with kynä, kirjahylly, sohvapöytä, etc. Definiteness or indefiniteness comes from context, word order, and sometimes from other words (like se = that/it), not from a specific article word.

Why does kirjahyllyn end in -n?

Kirjahyllyn is the genitive singular of kirjahylly (bookcase). The pattern is:

nominative: kirjahylly
genitive: kirjahyllyn

Here, kirjahyllyn is the word that vieressä (next to) “belongs to”. Vieressä is a postposition that requires its complement in the genitive:

pöydän vieressä = next to the table
talon vieressä = next to the house
kirjahyllyn vieressä = next to the bookcase

So the -n is there because vieressä governs the genitive case of the noun in front of it.

What exactly does vieressä mean, and what case is it?

Vieressä means next to, beside. It is related to the noun vieri (side, edge) and is in the inessive case:

base: vieri
inessive: vieressä (in/at the side → at the side of → next to)

In practice, you can treat kirjahyllyn vieressä as a fixed pattern meaning next to the bookcase, where:

kirjahyllyn = of the bookcase (genitive)
vieressä = at the side → next to (inessive/postposition)

What does the ending -llä in sohvapöydällä mean?

The ending -lla / -llä is the adessive case. One of its core meanings is location on or at a surface or area.

pöytä → pöydällä = on the table
sohvapöytä → sohvapöydällä = on the coffee table

So sohvapöydällä means on the coffee table.

Very roughly:

-ssa / -ssä (inessive) = in/inside something (e.g. laatikossa = in the drawer)
-lla / -llä (adessive) = on/at something (e.g. pöydällä = on the table)

Could you also say sohvapöydän päällä instead of sohvapöydällä?

Yes, but there is a nuance.

sohvapöydällä = on/at the coffee table, neutral and very common
sohvapöydän päällä = literally on top of the coffee table

In many everyday contexts, pöydällä and pöydän päällä both translate as on the table, but päällä focuses more explicitly on being on top of the upper surface. Pöydällä is slightly more general and can also feel a bit more idiomatic and shorter in ordinary speech.

In your sentence, sohvapöydällä is perfectly natural.

How do the two location phrases kirjahyllyn vieressä and sohvapöydällä work together?

They stack to give a more precise location:

kirjahyllyn vieressä = next to the bookcase
sohvapöydällä = on the coffee table

Together they express: on the coffee table, which is next to the bookcase. In English we usually say on the coffee table next to the bookcase; Finnish can simply put the two location expressions next to each other.

The structure is:

Viivain ja kynä (subject)
ovat (verb)
kirjahyllyn vieressä sohvapöydällä (combined location: next to the bookcase, on the coffee table)

Can I change the word order of the location phrases?

Yes. Word order with adverbials is fairly flexible. For example:

Viivain ja kynä ovat sohvapöydällä kirjahyllyn vieressä.
Sohvapöydällä kirjahyllyn vieressä ovat viivain ja kynä.

All of these can be understood as The ruler and the pen are on the coffee table next to the bookcase.

Differences are mostly about emphasis and flow, not grammar. The original order kirjahyllyn vieressä sohvapöydällä is natural and emphasizes first the relation to the bookcase, then the specific surface (the coffee table).