I am thinking of a five-digit number that is not divisible by three or four. If I increase each digit by one, I get a five-digit number that is divisible by three. If I decrease each digit by one, I get a five-digit number that is divisible by four. If I swap any two digits, the number decreases.

What number could I be thinking of? Find all possibilities.

(M. Mach)

Hint. Can you rule out certain digits in certain positions beforehand?
The property of digit swapping means that each digit of the thought number is greater than the following one, or smaller than the previous one. The property of digit enlargement and reduction means that the largest digit is less than 9 and the smallest digit is greater than 0. Therefore, in total,

1. the digit is less than 9 and greater than 4,
2. the digit is less than 8 and greater than 3,
3. the digit is less than 7 and greater than 2,
4. the digit is less than 6 and greater than 1,
5. the digit is less than 5 and greater than 0.

The property of divisibility by four means that the last two digits of the reduced number are divisible by four. Together with the previous conditions, this means that the last two digits of the thought number can be one of the following:

$$
51, \quad 43, \quad 31 .
$$

The property of divisibility by three means that the sum of the digits of the enlarged number is divisible by three, so the sum of the digits of the thought number gives a remainder of one when divided by three. All possible thought numbers are

$$
87643, \quad 76543, \quad 87631, \quad 86431, \quad 76531, \quad 65431 .
$$