Start to 02:40 - Folding paper

Lisa is challenged by the Guard to count the number of creases in a sheet of paper that's been folded twelve times. She finds, however, that she can only fold a standard sheet four or five times.

02:40 to 05:05 - The large sheet

The Guard has prepared a large sheet of paper for just this eventuality, which they manage to fold seven times (127 creases). It's still not enough. The Wizard, ever helpful, proffers a stick of chalk, suggesting it'll help.

05:05 to 05:58 - The Mathematical Encyclopaedia

The Encyclopaedia opens to page 196. Having introduced us to palindromes, it goes on to find arithmetical examples. For instance, take the number 87. Reverse it, and add the two numbers together. 87+78=165. Continue:

165+561=726. 726+627=1353. 1353+3531=4884. 4884 is palindromic. Starting from any number below 10 000 will yield a palindrome eventually, with one possible exception - the number 196. (At least, it's not thought to generate a palindrome. Only a few thousand reversals have been tried, so it's not proven.)

05:58 to 10:12 - The solution, and revenge

Lisa has worked out how many folds there would be, if she could indeed fold a sheet of paper a dozen times: 4095. She arrived at her answer by drawing up a table of results, and extrapolating. Through a twist of the regulations, she gets to set a puzzle for the Guard. The puzzle asks for one grain of rice for the first square on a chessboard, two for the second, four for the third, and so on. They head off in search of a chessboard.

10:12 to 11:12 - More on folds and creases

The Wizard suggests another investigation with folding paper: as well as counting the total number of creases, one can also count how many up and down creases there are.

11:12 to end - The chessboard

The Wizard joins Lisa and the Guard at the chessboard, where they're piling up the rice. By the time he arrives, they've already given up counting grains, and are weighing them instead. While this isn't completely accurate, it's close enough. They decide to skip forward a few squares, but as the Guard is trundling in more rice Lisa and the Wizard sneak off. We leave the hapless Guard as he tries to work out how many paddy fields' worth he'll need for the next square.