On the following image I additionally marked wards i use on regular basis in above average MMR (~3700).Works on AP mode on public matchmaking where counterwarding is less common.



Read full guide here for more detailed explanations about the marked colors and places.

For average skill games with low warding and no counterwards,you should focus mainly on red dots especially 2 hills near the runes with the priority of upper one in early game and lower one later when killing roshan becomes more likely.


Having at least one ward near the rune will eliminate the chance of
going the wrong side when timing runes
the hills will give you more vision than placing wards on the ground
easily prevent ganks since the river is most common way for ganking
seeing enemy team attacking roshan is huge advantage
also it reveals path near secret shop and neutral camp 




Wards should be placed as early as it gets and replaced just a bit before they expire(you can see the timer),it is the best if you can go grab a rune and place new ward at the same time so you don't lose much "lane time",if it is on opposite lane you could port there and combine it with a gank or push attempt.



There are 2 types of wards Observer wards and Sentry wards the ones you need are observer ones the other is used to reveal invisible enemies and is placed directly on the line near combat(pink)



If you want more advanced video guides you should check XV Rogues channel 

How do I ward properly in DotA 2?
Let $k(x) = x^2+2x+2$ be under $\mathbb Z_{11}[x]$. Determine if $k(x)$ is irreducible, and if so, determine if it is primitive.

Ok, I showed that $k(x)$ is irreducible since it it a quadratic and has no linear factors.

A field constructed with $k(x)$ would have order $11^2=121$...and so the multiplicative group would have $120$ elements. If $x$ generates this group under multiplication modulo $k(x)$, then I would know that $k(x)$ is primitive. However, I would have to calculate $x^1,x^2,...,x^{120}$ to show this...

I figured that it would be ridiculous for a question like this to result in $k(x)$ being primitive, but what if it were? Would I really have to calculate $x^1,x^2,...,x^{120}$ in order to demonstrate this? Is there a much more efficient way of solving a problem like this?
