There is no map that perfectly preserves both shape and size. This is because the Earth is a sphere, and it's impossible to flatten a sphere onto a flat surface without some distortion.

Here's why:

* Shape (Conformal): Maps that preserve shape are called conformal maps. They accurately represent angles and the relative shapes of features, but they may distort distances and areas. Examples include the Mercator projection.

* Size (Equal-Area): Maps that preserve area are called equal-area maps. They accurately represent the relative sizes of features, but they may distort shapes. Examples include the Gall-Peters projection.

So, what's the solution?

The best map for a particular purpose depends on what you're trying to show.

* If you need to accurately represent shapes, like for navigation, a conformal map is best.

* If you need to accurately represent areas, like for comparing the size of countries, an equal-area map is best.

Compromise projections: Some map projections aim to minimize distortion in both shape and size. These compromise projections, like the Robinson projection, are often used for general-purpose maps. However, they will always have some degree of distortion.