Who owes whom, in as few payments as possible SETTLE

The penny has to go somewhere, and it has to be named

Ten pounds between three people is 3.33 each, which is 9.99. The missing penny cannot be ignored: drop it and the books do not balance, round it up for everybody and the group is two pence out the other way.

The only correct answer is to give the remainder to specific people, by a rule that is the same every time, and to say who got it. Over one dinner that is a penny and nobody cares. Over a year of a shared house it is pounds, and because most spreadsheets round the same way every time, it is always the same person paying them, which is how a small unfairness becomes a real one without anybody noticing.

So the remainder is allocated deterministically here and shown. You can see who absorbed it, which means anybody can check, and the totals add up exactly to what was spent rather than to something close.

As few payments as possible

The other trap is the number of transfers. Six people with a tangle of shared costs produce a list of who owes whom that can run to a dozen payments, most of which cancel each other out: A owes B five pounds, B owes C five pounds, and C owes A five pounds is three transfers that achieve nothing.

What matters is each person's net position. Once everybody is reduced to a single number (up or down), the people who are down pay the people who are up, and the number of transfers is at most one less than the number of people involved. Six people go from a dozen payments to at most five.

That is not a cosmetic improvement. Every transfer is something somebody has to actually do, and a settlement with twelve steps does not get done: two people pay, the rest is forgotten, and the group is now in a worse state than before because the record says it was settled.

Getting the inputs right

Record who paid and who the cost was for, which are different and are the part people collapse. A taxi paid by one person for four of them is not a cost shared by everyone on the trip, and treating it as one is the commonest source of a disagreement that then gets argued as though it were about arithmetic.

Put costs in as they happen. Reconstructing a month of receipts from memory produces a settlement everybody has a reason to doubt, and the doubt is usually justified.

Agree the uneven splits explicitly and in advance: the bigger room, the person who does not drink, the one who arrived on the second night. Written down beforehand it is an arrangement; raised at settlement time it is a negotiation, and those go badly among people who are otherwise friends.

Nothing you type is uploaded, and there is no account, which for a record of who owes whom among a group of named people is the right default. Nearby: LEDGER keeps a cash book, RECEIPTS holds the paperwork behind each line, MONEY does the arithmetic on one-off sums, and DIGITS checks that a total is what it claims.