Janet and Fred Tompkins own four cats—Absalom, Booboo, Catnip and Dodo. When the Tompkinses go out, they leave the cats in the kitchen, which has a catflap so that the cats can go outside. This is the only exit and entrance. One day Janet and Fred left the cats, and all four quickly developed an urgent need to go outside.
"If Catnip and Absalom are indoors, they fight unless I'm there to stop it," said Dodo.
"Yes, and if you and Booboo are indoors, you fight unless I'm there to stop you," Absalom pointed out.
"Yes, and if you and Dodo are indoors, you fight unless I or Catnip are there to stop you," said Booboo.
"Well, if I'm left indoors with Catnip, she always causes a fight unless you or Absalom are present," replied Dodo.
"If I'm left indoors alone, I won't go out at all," said Absalom.
"The same goes for me," said Booboo.
When the Tompkinses got home, all the cats were out, but none of them had fought.
How did the cats do that?
(Cats can return to the kitchen if necessary. Only one cat at a time can use the catflap.)
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The answer
C goes out, D goes out, A goes out, C comes back in, B goes out, C goes out.
There are 16 possibilities for which cats are in the kitchen: ABCD, ABC, ABD, and so on until none are present—call this *. The arrow > denotes a move. (Ignore trivial variants, in which the same cat goes out and comes in again.)
The following combinations have been ruled out: ABC, AC, BD, BCD, AD and CD; also the moves A > * and B >*.
So ABCD > ACD or ABD.
As ACD > AC, AD, or CD, all of which are ruled out, then it is ABCD > ABD.
Since ABD > AD and ABD > BD are ruled out, it must be ABD > AB.
But neither AB > A or AB > B can lead to *. Another cat must come back in.
So AB > B, because with AB > A, AC and AD are ruled out. B > BD is ruled out, so B > BC. Then BC > C > *.
The winner was Ian McGuffog from Exmouth
