"A mi székeink újak és szépek."
Translation:Our chairs are new and beautiful.
So the rule isn't simply that the -i you add when making a possessed noun plural is added to the end? If that we the case, "our chairs" should be a mi székünki. Is it that it gets placed after the last vowel? (And then I guess ü becomes e to make it sound nicer...?)
Fortunately, it's a lot simpler. :D
If the noun ends with a consonant, you take the third person singular possessive singular (not sure about the English word order here), then + i to put it into plural, then conjugate appropriately.
szék = chair
széke = its chair
székei = its chairs
székeim/d/nk/tek/k = my/your/our/your/their chairs
Note that you don't go through the third person singular possessive singular if the noun ends with a vowel, although the intermediate step is still the third person singular possessive plural after all.
kutya = dog
kutyája = its dog
but! kutyái = its dogs
kutyáim/d/nk/tok/k = my/your/our/your/their dogs