N-body problem and the Moon

The Earth Moon Sun gravitational relationship is a n-body problem. It is not a series of 2 body problems. This problem has stumped me since I was 11. I did not realize then that this was considered an impossible problem to solve. What really frustrates me is that the universe solves this problem so effortlessly. Its like an elegant ballet.

The general 2 body problem is F = -m₁m₂G / r^2

The problem I have with many n-body attempts continue the linear solution. Solve one set of 2 body problems and continue this until particle positions/velocities/... all have been solve. The problem here is that it never works.

Here is the reason why. Most of the solutions always return to a series of 2-d problems. The universe is at least 4-d. And Gravity is a function of density.

The best/most correct/closest solution I have found is that by Peter Duffet-Smith. His book "Practical Astronomy with your Calculator" has follow me most of my life. In this book he describes the moon's orbit.
"A celestial observer viewing the solar system from a great distance would not, however see the Moon making loops in space about the Earth. Rather, he would describe the situation by saying that the Moon is in orbit around the Sun, as is the Earth, and that the effect of the Earth's influence is to make the Moon's orbit wiggle a little as the relative positions of the Earth and Moon change. This is because the Sun's Gravitation force on the Moon is much greater than that of the Earth, even though the latter is nearer. "

This is a brilliant book. He should get a Nobel Prize just for that paragraph. And he is correct. Please buy his book. He even has great calculations to prove his theory. He needs to be planted firmly in the lexicon of great physicists.