Fields fade fast

Far from a small magnet, its field falls as 1/r³, where r is the distance from the magnet's centre. Double the distance and B drops to (1/2)³ = 1/8. Triple it and B drops to 1/27.

Why so fast? A magnet is a dipole: an N and an S pole close together. Far away, their two fields almost cancel, and what is left shrinks faster than the 1/r² you get from a single source. That is why a magnet that grips steel at contact barely moves a compass half a metre away.

Did you reach 1/8 a little before 16 cm? The 1/r³ rule is exact only when you are far away compared with the magnet's size. This bar is 6 cm long and you started only 8 cm from its centre, so the near pole still matters. Close up, the field falls even faster than the ideal rule.

Try going further out: the farther you are, the better each doubling gives exactly 1/8.