Henry Ernest Dudeney/Modern Puzzles/81 - Pussy and the Mouse

by : $81$

 * Pussy and the Mouse
 * "There's a mouse in one of these barrels," said the dog.
 * "Which barrel?" asked the cat.
 * "Why, the five hundredth barrel."
 * "What do you mean, the five hundredth? There are only five barrels in all."
 * "It's the five hundredth if you count backwards and forwards this way."
 * And the dog explained that if you count like this:

1  2   3   4   5 9   8   7   6    10  11  12  13
 * so that the seventh barrel would be the one marked $3$ and the twelfth barrel the one numbered $4$.

The story goes on laboriously to its inevitable conclusion that the mouse escapes before the cat has finished counting, until:


 * Now, which was the five hundredth barrel?
 * Can you find a quick way of arriving at the answer without making the actual count?