I remember discovering that pi x 10^7 is very close to the number of seconds in a year while at uni.

One of my tutors was convinced this had to be more than coincidence, but I always figured it was just chance and a nice but sometimes useful shortcut...

I always liked the fact that 10! (10 factorial) is exactly the number of seconds in six weeks.

  6 weeks * 7 days * 24 hours * 60 minutes * 60 seconds:

    6 * 7 * 24 * 60 * 60
  = 6 * 7 * (3 * 8) * (4 * 5 * 3) * (3 * 2 * 10)
  = 6 * 7 * 3 * 8 * 4 * 5 * (3 * 3) * 2 * 10
  = 6 * 7 * 3 * 8 * 4 * 5 * 9 * 2 * 10
  = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10

You might be able to send someone down an amusing (to observers) rabbit hole of wrongness by telling them it is not exact because Earth’s orbit is not perfectly circular.

It cannot be anything but coincidence. While 365.25 days in a year is physics, a day consisting of 86,400 seconds is an entirely arbitrary human construct.

Get enough numbers, accept wide error bars, and some of them are going to overlap.