Block puzzle

Here is the puzzle:

Now obviously it is not the most difficult puzzle ever but it is fairly tricky and not the kind you can glance at and tell the solution. So let us find out a method to solve it.

Throughout this article, 11, 12,21,22 will represent the smallest blocks, the block in the middle, the blocks on either side of the biggest block, and the biggest block respectively as shown in the puzzle before the blocks are moved. 1rd, 1ru, 1cl, 1cr will represent ‘go down by one row’, ‘go up by one row’, ‘move left by one column’ and ‘move right by one column’ respectively. The blocks are placed on, as we call it, a 5X4 rectangle.

First, the clues that we can get from the picture.

1. There will be a time when 22 will have to cross 12. Then no other block can move from one side of the separation line that 22 and 12 create to the other.

2. While crossing, when 22 goes one row down 12 must go one row up.

3. While crossing, all 21s must be on one side of the separation line, which implies that 12 must be in the bottom row. Look at the picture with clue 1. 22 occupies 2 rows and 12 occupies 1. If 22 takes the first step down, 12 takes the next step up and then 22 takes the next step down. If 12 takes the first step up, then it’s 12 1ru,22 1rd,12 1ru. Any one of them has to move 2 rows which is impossible if one side of the separation line has even one 21.

4. As 22 goes one row down, two 11s must take the place it vacates.

5. Hence when 22 takes the first step down it must do so along the 1st or the 4th column.

6. So clearly, when 22 takes the first step down, two 21s must be right under it, because other two options are impossible.

The first one is impossible because in the previous position 22 must have been one row up, so the two 11s were on top of the two 21s and… you’re stuck. The second one is impossible for similar reason, just take the only possible steps backward. Remember, 22 can’t go down a row because it is its first step down and we are retracing the steps.

So the only possibility that remains is  

Retrace the path to find out how we can come here, it is pretty much one way.

So now, where to go from here? Clearly we have to move 22, otherwise our target of getting all 21s on top cannot be achieved. To make room to move 22, move the 11 of the 3rd row and 3rd column to the top row and then move 22. Now clearly you’ll have to move the bottom 21 up next to the top corner 11, which will leave you room to move its adjacent 21 to the place it vacates. Of course the next move should be of the 11 in the 4th row. Now you can move 22 one row down and the two 21s of the last column up to allow the 22 to move on top of 12. Now move a 11 down and the rest is pretty simple.