Nathaniel
During the most of the events of The Amulet of Samarkand, Nathaniel is twelve:
The article was on page two, sandwiched between the Employment
Ministry’s monthly request for new apprentices and a short report from
the Italian campaign. It was three columns in length. It noted with
regret the deaths in a severe house fire of the Internal Affairs
Minister Arthur Underwood and his wife, Martha. The blaze had started
at approximately 10:15 P.M. and had only been fully extinguished by
fire crews and emergency service magicians three hours later, by which
time the whole building had been gutted. Two neighboring houses had
been badly affected, and their occupants evacuated to safety. The
cause of the fire was unknown, but police were keen to interview Mr.
Underwood’s apprentice, John Mandrake, aged twelve, whose body had not
been recovered.
—The Amulet of Samarkand
During The Golem’s Eye, he is fourteen:
More than two years had passed since the time of Lovelace’s rebellion,
and the sudden emergence of Nathaniel into the elite. By now, he was
fourteen years old, taller by a head than when he had returned the
Amulet of Samarkand to the protective custody of a grateful
government; bulkier, too, but still lean-framed, with his dark hair
hanging long and shaggy around his face after the fashion of the day.
—The Golem’s Eye
And finally, during Ptolemy’s Gate, he is seventeen:
John Mandrake was an attractive young man, and the scent of power hung
about him, sweet and intoxicating, like honeysuckle in the evening
air. He was of medium height, slender of body, and swift and confident
in action. His pale, slim face presented an intriguing paradox,
combining extreme youth—he was still only seventeen years old—with
experience and authority.
—Ptolemy’s Gate
Well, that was easy.
Kitty
Figuring out Kitty’s age is a little harder. She seems to be somewhat older than Nathaniel. During the account of how she joined the Resistance, she is mentioned as being thirteen:
She and Nicholas, a stocky young man with a brooding manner, had
worked for Mr. Pennyfeather since they were children. The rest of the
company were younger; no one older than eighteen. Kitty and Stanley,
both thirteen, were the youngest of all.
—The Golem’s Eye
She’s mentioned as being fifteen some time later:
Time passed. Kitty left school at the standard age of fifteen.
—The Golem’s Eye
So she was probably thirteen during The Amulet of Samarkand, but she could conceivably have been fourteen. Her age during the other books is more or less a simple matter of incrementing: if she was about thirteen during the first book, for example, she would have been about 18 during the last one.