A puzzle by Henry Dudeney:

‘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 by the five-hundredth? There are only five barrels in all.’

‘It’s the five-hundredth if you count backwards and forwards in this way.’

And the dog explained that 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.

‘That will take some time,’ said the cat, and she began a laborious count. Several times she made a slip, and had to begin again.

‘Rats!’ exclaimed the dog. ‘Hurry up or you will be too late!’

‘Confound you! You’ve put me out again, and I must make a fresh start.’

Meanwhile the mouse, overhearing the conversation, was working madly at enlarging a hole, and just succeeded in escaping as the cat leapt into the correct barrel.

‘I knew you would lose it,’ said the dog. ‘Your education has been sadly neglected. A certain amount of arithmetic is necessary to every cat, as it is to every dog. Bless me! Even some snakes are adders!’

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