I cantell you the perfect books,for i have had quite a search indeed!

What a great start!

Hear is a list for building deep understanding and problemsolving ability.
note: Some high school courses,and teachers are incompetent,beaware. Maths is great fun and stimulation,but at the worst its drill
and factual learning.

Dover publishershaveexellent cheap stuff (like thefirst two books)
a cataloge is free.
iANSTUART ANDMARTIN GARDNER are great.

1.The moscow puzzles by Boris A. Kordemsky

This is mathematical puzzlesvery easy to very hard, needing no actual knowldge really- doeshelp,butit is common sense which it workson, to build problemsolvingandgaininginsits through puzzles.
MY FAV. CHAPTER IS "SKILL WILL FIND ITS APPLICATION EVERYWHERE"
(dam caps!)
Martin gardneris good too.

2. Amusements in mathematics by Dudney
bit like above, more puzzles,but 1. is better,i think.

3.what is mathematics by courant and robbins.
bit advanced at some parts, but i think important for whole high school.
4. some ofthe juniour mathematical olympics handbooks.

-imaginationlogic@lexicon.net