Can you solve it? Intrigue at the pet hotel | Mathematics

Today’s brain-manglers come from Tadao Kitazawa, a prolific creator of maths and logic puzzles from Japan. Over the last two decades he has introduced many original ideas, and given new twists to established genres.

Puzzles, he writes, are about having fun with limited information. He tries to devise problems that look complicated at first sight, but that are very simple once you are on the right track. Are you feline lucky?

1. The Pet Hotel

In the Pet Hotel, the rooms are numbered 1 to 5, in that order. Each room can accommodate one animal, and has its own light. At night, an animal who is nervous leaves the light on. An animal who is not nervous turns the light off. Each of the rooms 1 to 5 are always occupied by either a dog or a cat, and everyone checks out after a night.

a) On Saturday night, a dog is nervous if and only if there are cats in both adjacent rooms. A cat is nervous if and only if there is a dog in at least one adjacent room. It is observed that four rooms remain lit. How many cats are there at the Pet Hotel?

b) On Sunday night, a dog is nervous if and only if there are other dogs in both adjacent rooms. A cat is nervous if and only if there is another cat in at least one adjacent room. It is observed that only one room remains lit. How many cats are there at the Pet Hotel?

2. Shaken, not bumped

Among six children, each handshake is between a boy and a girl. Each of four children shakes hands with exactly two others. Each of the other two shakes hands with exactly three others. Do these children shake hands with each other?

3. I should be so lucky

Three girls, Akari, Sakura and Yui, are each given a positive whole number, which they keep secret from each other. They are told the sum of the numbers is 12. A girl is considered “lucky” if she has the highest number. It is possible that one, two or all three girls are “lucky”.

Akari says: “I don’t know who is lucky.”

Sakura says: “I still don’t know who is lucky.”

Yui says: “I still don’t know who is lucky.”

Akari says: “Now I know who is lucky!”

Who is lucky?

Purr-lease NO SPOILERS. I’ll be back at 5pm UK with the solutions.

UPDATE: Solutions can be read here

Kitazawa, aged 59, lives in Nagano, where he writes local history books as well as submitting puzzles to Konwakai News, the monthly bulletin of the Academy of Recreational Mathematics of Japan.

A book of his puzzles Arithmetical, Geometrical and Combinatorial Puzzles from Japan has just been published by the Mathematical Association of America and the American Mathematical Society. There’s a huge amount in there that will delight puzzle fans. The problems above are taken from that book, slightly modified to fit the format of a column.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

Football School Greatest Ever Quiz Book

I’m the author of several books of puzzles, and also the children’s book series Football School. The latest instalment, The Greatest Ever Quiz Book, is out on Thursday. You can pre-order at the Guardian Bookshop, and anywhere else you buy books.

I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.

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