Three Short Logical Thinking Puzzles

The First Puzzle:
In the ancient days, came a man from countryside to a small township for living better. But upon his utmost efforts, he failed and in a great depression. It was said that he had been fool and had disappeared. Several days later, someone found him in the empty house but he was dead. When a detective was called at the spot, he saw that the dead man still hanging there by a rope around his neck, and the other side of the rope tightened with the girders of the ceilings. On the ground there was still a pool of water. The house was totally empty and the girders were very high that the dead man could not reach the girders from the ground even plus the rope, but still could not get over there. Anyway, the detective at last got the answer that it was a suicide. But the question is how did he do that?

The Second Puzzle:
Once upon a time, there was a woman visiting her friend. She had to take a rest near a small lake in the middle of the way. In the lake centre, it had a tiny island and on it there was a tree. The horse she travelled with could not be tightened along the side of the lake. Neither could she swim. But she only had a very very long rope. At last, she tightened horse and went to a rest. Will you tell me how she did that?

The Third Puzzle:
There was an old man long time ago. He marched a long way to reach his destination. In the middle of the road, he was stopped by a long and wide river. There was no bridge or boat whatever tools to across that river. The only available thing left for him was his neck scarf. Finally, he successfully got to the other side and reached his destination. How did he go cross the river?
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By Lingyun