Start to 01:45 - Setting the scene

Lisa, with nothing better to do, persuades the Guard to try her wit and cunning: they decamp to the Number Grid, where they have to 'walk the multiple paths, but never take the first step'.

01:45 to 05:45 - Priming the grid

With the aid of an obscure - and somewhat embarrassing - cheerleading audition from the Wizard, Lisa works out that they must cross off all the numbers on the grid which are multiples of 2. They must, however, 'never take the first step' - which she takes to mean leaving the number 2 itself in place. Having crossed off multiples of 2, she moves on to multiples of 3, and notes that multiples of 4 have already been covered. The Guard guesses that, in time, they'll cross off the entire grid, but Lisa isn't so sure.

05:45 to 06:30 - The Mathematical Encyclopaedia

The Encyclopaedia opens to page 'not one'. Having defined prime numbers, it describes the difficulty presented by the number 1, and notes that 1 is by definition not a prime number.

06:30 to 09:55 - The second problem

Having worked out that the procedure left behind the prime numbers, Lisa persuades the Guard to give her another problem. This time, she must remove all the squares which can be made by summing consecutive numbers. For example, 3 can be removed, since 1+2=3. So can 18: 3+4+5+6=18.

09:55 to 10:55 - Marin Mersenne and his primes

The Wizard introduces us to Marin Mersenne, a French mathematician who was interested in very large prime numbers. He came up with a simple way of generating very large potential primes, a method which is still of use today.

10:55 to End - Powers of two

Lisa is somewhat stumped. The squares left don't follow any pattern she can discern, and she's not even sure which ones she should be able to block off. Enter the Wizard, whose anecdote about Noah's Ark spurs her into noticing that most of the remaining numbers are powers of two. Quickly, the heroes polish off the remainder, at which point the Wizard recruits them for another scheme of his...