Start with a tab at the tavern

You and eleven friends drink at the same tavern every week. Nobody pays on the spot — the keeper writes it all in a book: Ana owes 3, Bruno owes 5, Ana paid 3 back.

The book works. But it has one weakness, and it is not the arithmetic: the keeper is the only one holding it. If a page gets rewritten one quiet night, there is nothing to compare it against.

Everything in this chapter comes from fixing that single weakness. No mathematics required — just a better arrangement of the book.