# The Battle of the Books

Andrew the Assured, Brian the Big, Casey the Careful, Dootles the Dreamy and Edwina the Eager are all members of a reading club, ‘The Burping Bookworms’. They have a bizarre rule in the club: for each 5-person group, they must read 5 books in week, but in a special way–after they finish reading each book they must exchange them within the group, so that after several switches, each of them could finish reading the same 5 books at the same time. (Suppose they could all read at the same speed and swap books at the same time.)

After a pleasurable week of switching-and-reading, Casey the Careful was chatting with her friend, ‘3M’, Mazy the Mathematics Maniac.

Suddenly, Casey piped up, ‘You’re a math frenzy, Mazy, and you know about our ‘Burping Bookworms’ reading club, so I’ll give you a puzzle to solve.’

Casey took out a sheet of paper on which she jotted down facts:

1. The last book Andrew read was the second book Brian read;
2. The last book Casey read was the fourth book Brian read;
3. The second book Casey read was handed to her by Andrew;
4. The last book Dootles read was the third book Casey read;
5. The fourth book Brian read was the third book Edwina read;
6. The third book Dootles read was the book Casey had borrowed from the library (and read) in the first place.

‘Could you figure out the sequence in which we read these 5 books?’

Well, as a matter fact, Mazy was a real math maniac, and in a short time she was proudly handing her work over to Casey. Can you?

This problem was sent to WTM by Wanting Zhong, our friend from China.

# Arthur’s Arithmogons

Arthur has invented a new puzzle to show his friends. He calls it: Arithmogons.

He draws the following diagram first.

Then he puts any three integers he likes in the circles. In the squares he puts the sum of whatever two integers appear on the ends of the corresponding side.

Before showing this to a friend, he erases the integers in the circles. He tells his friends to find the circle numbers, knowing only those in the boxes.

Here is a sample of Arthur’s Arithmogons for you to solve.

EXTRA: while it’s possible to solve an Arithmogon puzzle by guess-&-check, it would be much more efficient to develop, and prove, a logical step-by-step approach. Such a method exists. Please find it and prove why it works.