A cat is at the bottom of a 20 foot well. (Don't worry, he's got lots of food, water, catnip and spitty cat toys down there.)
Each day, he climbs up 5 feet...

And each night, he slides down 4 feet.

How many days will it take him to reach the top of the well?

Either 20 or 4 if your classing days and nights as separate and trying to trick us through the wording?

Day 1 up 5 ft, Night 1 down 4ft = 1ft
Day 2 up 5ft etc etc = 2ft

By Day 15 the cat hits 19ft and goes down 4 ft = 15ft

The cat climbs another 5ft the next day to reach 20ft (the top of the well) so it's 16 days.

Big tick gold star

The cat won't have the manual dexterity or strength to carry up the food he needs from the bottom of the well with him & will be caught in an endless cycle of climbing up the well & coming back down for grub to avoid starvation, until of course, he runs out of food.