my very first practice of basic checkmate and I'm puzzled with computer answer

My apologies in advance for an elaborate question from a complete chess newbie. Not sure how to include diagram yet but I'll describe the setup. This is my very first time practicing basic checkmate; the board features only one opposing piece ( black Kd6 ). There are three white pieces: Ke3 Qa1 Rh1. Goal of practice is to use Q and R to mate ( three moves would be perfect mate ). Here is how I proceeded:

1) Rh6+ Kc5 //

2) Qa3+Kc4 //

3) Rh5 //

At this point computer said the game would be a draw and that I blundered ( it said the best move would have been Rb6 ). Now for the life of me, I can't see how the position of black Kc4 with the aforementioned configuration ( using my third move Rh5 ) would allow for anything but a mate. Black K can't move anywhere on 5th rank due to position of Rh5. Black K can't move anywhere on 4th rank due to position of Qa3 and white Ke3. And Black K can't move anywhere on 3rd rank due to same position. Would be grateful to know what I'm overlooking. Thanks very much. BTW, I've tried to insert a board with initial position but I'm not sure it will work. 

"Black K can't move anywhere...".

Exactly. It's a stalemate.

https://en.wikipedia.org/wiki/Stalemate

Thanks for the quick response! I assume the computer suggestion of alternate move was to improve the chance of an actual mate rather than stalemate. And I also assume that my initial moves were not sufficient to produce a mate in 3 so the computer was just responding to my ( less than perfect moves ) with the goal of victory. Much more study ( a lifetime worth ) remains. Thanks again!

A stalemate is a draw. If you make a move, that does not put your opponent in check AND ALSO does not allow your opponent to make any legal moves - it's a draw.

The simplest checkmate pattern in a position like this is probably rolling the queen and rook up both ends of the board, eg. 1. Rh5, king anywhere... 2. Qa6... king anywhere... 3. rh7.. king anywhere and finally 4. Qa8# 
If the king manages to approach the rook on H7, via say, G8.. then rather than 4. Qa8+, instead play Qa7, and then whereever the king moves on the 8th rank, just move the queen directly infront of the king, and it's mate (as the queen will be backed up by the rook already on h7). 

Thank you; I will study your pattern. 

Put someone in check every move and you can't stalemate them

The aim of chess is to checkmate your opponent, and this can be in any number of moves.  From the starting position shown I would aim to drive the king towards the back rank and checkmate there. 

In this example the King start to attack the rook.  I elect to move my rook to the other side of the board so in this example it takes 5 moves.  I simply do not have to think about it.  

Thanks for the detailed reply; it will take me a long time to get to the point where I can make those moves without thinking. Also, I was getting hung up on trying to make the mate in 3 moves ( which the computer said was possible ). I ( quite literally ) ended up in stalemate instead. 

Another possible idea

The idea is a repetition of a run down if you observe the following carefully.

Personally, I would perform a checkmate I am much more comfortable in than to try and launch a checkmate in as short as possible. Most players would agree with me that an easy mate in 7 is much preferable as compared to a very challenging mate in 6.

Take the following position, where there are lots of easy mates in 3 but only one mate in 2.

With White to move, only one move allows a mate after any of Black's moves in the following turn. But I would rather play 1. Rgd4+ and win the knight because it is just so...automatic for me.

Very handy pattern for bullet games.

the computer asks for 3 moves, which are difficult to find. in a real game, don't worry how many moves it takes to make a mate, just do it

It took me a few minutes to find the mate in 3, and yes I had to think about it.  I simply would not bother to think about it in a real game, and getting into a stalemate is easy to do if you make it complicated.  

Please give me a link to the original puzzle, and I'll report it as not being for beginners.  

I will explain that the key move is Qa4 because it blocks the b5 square for the black king, and forces the black king to Kd5, where your own white king is now close enough to support the queen.   

No human player would think about a mate in three from the original position.  As said Rh5 traps the king in the back three rows, and whilst it might take a couple of moves longer, over the board there would be no thinking, and it would in reality be a lot quicker.  

Thanks to all for such detailed responses; very kind of you. I will study tomorrow from the original position, taking into account the various suggestions. 

It was actually on another platform and not a puzzle per se, but an area which is titled "Piece Checkmates I: basic checkmates". The suggestion was, based on original position, to use queen and rook to restrict the king and then checkmate. Then they added 'mate in 3 if played perfectly'. As I now see ( from your advice and that of others ), being distracted by the 'mate in 3' caused me to 'make the perfect the enemy of the good' . BTW, I will check out the beginner section of Chess.com ; I've been searching a few other sites to see which one is the best fit. You folks are certainly quite helpful here. Best Wishes, Ian 

Yeah, I found the mate-in-3, but I had to think about it. In an actual game, I probably wouldn't have bothered trying for the mathematically shortest win; I would have just played something like blowerd's post above.

I think staircases are more normal for chess players. Most players won't bother to find the mate in 3.

sometimes you can be so far ahead in material you don't win a game because of stalemate. just try to be really careful when you've got an advantage like this. practice taking away squares from the king but putting your pieces as far away from the opposing king as possible. your king and queen suck away loads of squares from fleeing monarchs at the end of games!

I had a huge material advantage against a tricky young rival, and realized my only danger was stalemate.  I resolved to check each proposed move TWICE to make sure it wasn't stalemate, because I knew I was tired and an error would be easy.

That second safety check was the only thing that kept me from stalemating her.  Don't know how I missed it the first time!  But it shows the importance of watching out for this.

In real games you can sometimes make your job easier by refusing to take the opponent's last pawn, so he has something to move and can't be stalemated. (Just don't let it queen!)