[pdurrant]

Towels never need washing.*

*

Spoiler:

Axiom 1: When two things touch, the cleaner one becomes slightly more dirty, and the dirtier one become slightly more clean.
Axiom 2: The cleanest thing in a house is the most recently washed thing.

Deduction 1 from axiom 2: just after taking a wash in a shower or a bath, the just-washed person is the cleanest thing in the house
Deduction 2 from axiom 1 and deduction1: when that just-washed person gets dry using a towel, the towel become slightly cleaner and the person become slightly dirtier

Conclusion: Towels get cleaner every time they are used, and so never need washing.