Why a GPS fix needs a spare satellite
A GPS satellite broadcasts its position and the exact moment it sent the signal, timed by an atomic clock synchronised to within a few nanoseconds of every other satellite's. Your receiver notes when the signal arrived and multiplies the gap by the speed of light to get a range.
That range is a lie, and a specific one: the receiver's own clock is a cheap quartz crystal, not an atomic one, so it is off from true time by some unknown amount b . Every range it computes is too long or too short by exactly c·b — the same error, in the same direction in time, on every satellite it hears at once.
rangemeasured = rangetrue + c·b
Treat one unknown clock error as a fourth variable sitting next to your three position coordinates, and the trick becomes obvious: find the one value of b that makes every satellite's circle (or, zoomed in close enough, line) agree on a single point. Two unknowns plus a shared bias need three measurements to pin down in a plane; in the real three-dimensional world it's four unknowns — x, y, z and b — which is exactly why a phone needs four satellites in view, not three. The fourth one isn't for redundancy. It's what lets the receiver get away with a clock that costs a few cents.
Why some satellites help more than others
The sharpness of the fix depends on how spread out the satellites are, not just how many there are. Lines crossing at steep angles pin a point down tightly; lines that are nearly parallel can slide a long way past each other for almost no change in how well they "agree." Surveyors call this dilution of precision , and it's why a receiver boxed in by tall buildings — seeing only satellites bunched near one patch of sky — reports a position that wanders far more than the same receiver standing in an open field.
The scene above is schematic, not to real orbital scale: a GPS satellite is about 20,200 km up, so near your receiver its circle of equal range is, for all practical purposes, a straight line. Drawing it as one from the start is the same approximation GPS receivers themselves use.