Тройка: three of something
If numbers can be said to have personalities, три (three) is deeply conflicted. It is highly spiritual but sometimes very naughty. It is a very small amount of something, unless it is more than you can imagine. It is average and in between, but at the same time it is exalted and magical. If три were a person, he would be on drugs.
Of course, три is just the number three, helpfully defined as more than two and less than four. Три describes three units of something, like три часа ночи (3 a.m.; literally and properly "at night" in Russian).
Три is the grade you get in school when you aren't at all good, but not that bad. Три is a "C," the average mark. If you get this grade often, you might call them тройки, like old friends. If you get a great many of them, other people might call you троечник or троечница (C-student).
Трёшка used to be slang for a three-ruble note. Now it can be slang for various threes, like a three-room apartment: Мы хотим купить трёшку недалеко от метро (We want to buy a three-room apartment near a metro station).
Or a three-star hotel: Хорошая трёшка, очень удачное место расположения и рядом с отличным кафе (A solid three-star, in a very good location with a great cafe next door).
But три isn't always humdrum. Sometimes it is exalted, in a secular and superstitious sort of way. Бог любит троицу (God loves a Trinity), Russians will say when they are expecting a third event to happen — or when they hope the third time for something will be the charm.
Бог триедин (God is triune), they say, три — это любимое Божье число, будем надеяться, что в третий раз обязательно получится (three is God's favorite number, and so we hope that it will work out the third time for sure).
Because it is a godly number, Russians kiss each other on the cheek три раза (three times). And because it's got a bit of magic in it, Russians spit три раза over their left shoulder to ward off evil.
But sometimes три is in the gutter. Послать на три буквы (literally to send someone to three letters) is a very strong way of telling someone where to go — and it's nowhere good.
In some expressions, три means a lot. Пришли соседи и ели в три горла (The neighbors came over and ate us out of house and home). Муж ушёл, а она повалилась на диван и зарыдала в три ручья (Her husband walked out, and she threw herself on the couch and cried her eyes out).
In other expressions, три is very little of something. It might be a short distance. Они живут в трёх шагах отсюда (They live just a few steps from here). Or very few of something: Заблудиться в трёх соснах is to get lost in three pine trees; it is said of somebody with a bad sense of direction.
But it can also refer to someone so dim-witted they couldn't think their way out of a paper bag. Он превратил пустяковое дело в головоломку и заблудился в трёх соснах (He turned a molehill of a problem into a mountainous mind-boggler and then couldn't think his way through it).
А решение проблемы было всего в трёх шагах … (But the solution to the problem was within reach …).