Three logical tasks, solved by the supercompiler JScp    

russian.htm - russian text,

fish - task of A.Einstein,

agent007 - task about the agents,

recnumb -  task about recursive numbers.

The task 1. The task of A.Einstein   fish

There are 5 houses in 5 different colors.  In each house lives a man with a different nationality.  The 5 owners drink a certain type of beverage, smoke a certain brand of cigar, and keep a certain pet.  No owners have the same pet, smoke the same brand of cigar or drink the same beverage.

The question is: "Who owns the fish?"

Hints:

• The Brit lives in the red house.
• The Swede keeps dogs as pets.
• The Dane drinks tea.
• The green house is on the left of the white house.
• The green house's owner drinks coffee.
• The person who smokes Pall Mall rears birds.
• The owner of the yellow house smokes Dunhill.
• The man living in the center house drinks milk.
• The Norwegian lives in the first house.
• The man who smokes Blends lives next to the one who keeps cats.
• The man who keeps the horse lives next to the man who smokes Dunhill.
• The owner who smokes Bluemasters drinks beer.
• The German smokes Prince.
• The Norwegian lives next to the blue house.
• The man who smokes Blends has a neighbor who drinks water.

The task 2. The task about the agents. agent007

The Agent 001 watches the agent, who watches the Agent 002.

The Agent 002 watches the agent, who watches the Agent 003.

The Agent 003 watches the agent, who watches the Agent 004.

The Agent 004 watches the agent, who watches the Agent 005.

The Agent 005 watches the agent, who watches the Agent 006.

The Agent 006 watches the agent, who watches the Agent 007.

The Agent 007 watches the agent, who watches the Agent 001.

All went normally until the Agent 008 appeared. In this case it became impossible for the chief to make up the similar scheme of relative shadowing for eight agents.

1. What scheme of relative shadowing was at first (whom each agent watches)?

2. Why is the 8 agents scheme impossible?

The task 3. The task about recursive numbers.  recnumb

Find such a 10-digit number, in which the first digit is equal to the amount of zeros in this number. The second digit is equal to the amount of units in this number, and so on. The tenth digit equals to the amount of nines in this number.

For example, the number 6 210 001 000 . It has 6 zeros, 2 units, 1 digit two, 1 digit six. There is no other digits.

The task is extended for n-digit  numbers.

The answer to this task is quite curious.

n =2  - no solutions,

n = 3 - no solutions,

n = 4  - two solutions - 1210 and 2020,

n = 5  - one solution - 21200,

n = 6  - no solutions,

n = 7  - one solution - 3211000,

n > 7 - one solution -

n-4 , 2 , 1 , 0 , ... , 0 , 1 , 0 , 0 , 0 .

fish - task of A.Einstein, agent007 - task about the agents, recnumb -  task about recursive numbers.