Cats in Boxes (Not Schrödinger)

[P4 Maths and Up: Logic Puzzle]

Kept you waiting, huh?

Here's an interesting puzzle you may, or may not, have seen before.

The hiding cat puzzle ends with a fairly simple answer, to do a sweeping search of the boxes (leave the last two boxes unchecked, and check all other boxes row by row) Of course, this puzzle only really works because of a few limitations:

1) The cat only moves at night.

2) The cat is forced to obey the rules and move to a different box ever night.

But anyone who's ever taken care of a cat in proper would find that inaccurate. They're unpredictable, they're mischievous and they like playing as much as they love lazing around. Their personalities can be quiet and attention-demanding, or they could be tempestuous, clawing and damaging curtains and couches while scattering mud all over the house without restraint.

So, here's a small puzzle for you: Let us assume that the cat is smarter than it looks and is able to dig tunnels between the boxes, but it doesn't stay in the tunnels, because, well, the soil stinks of ammonia, methane and fertiliser extracted from the cow's crapper. The cat is also fairly mischievous and wants to play, to spend time with you, so it will actively make itself as hard to find as possible; staying in a box you have not checked, dodging you in the tunnels.

This would make the cat much harder to find and, frankly, make the task of catching it feel impossible, so there's a caveat; the tunnels that the cat digs may only connect between two boxes. The tunnels cannot merge, because the soil will loosen too much and collapse the tunnels, and the barmy cat instinctively knows & avoids this; after all, if it got stuck, it wouldn't be able to play with you.

To summarise:

A) Cat wants to play, and will actively avoid you as long as possible

B) A box can have multiple tunnels to-and-fro, and the cat uses these tunnels to extend its hide-and-seek with you

C) Every tunnel can only connect between two boxes

D) The cat doesn't want to stay in the smelly tunnels, so it can't hide inside of them.

Can you come up with a strategy to catch the cat? (Answer revealed sometime in the future)