1. Problem statement: Roz is receiving overnight care and her kids fronted payments and there have been transfers among the kids; compute who owes whom and how much Roz owes so everyone is fully reimbursed. 2. Required data: provide a list of participants and a list of transactions where each transaction is who paid, for whom, and the amount. 3. Notation: index persons by i = 1,...,n with Roz included and let $M_{i,j}$ be the amount person i paid for person j. 4. Formula used: compute each person's net balance by $$B_i = \sum_{j} M_{i,j} - \sum_{j} M_{j,i}$$ 5. Rule: if $B_i>0$ then others owe person i amount $B_i$; if $B_i<0$ then person i owes others amount $-B_i$. 6. Settlement algorithm: list creditors ($B_i>0$) and debtors ($B_i<0$), then repeatedly match the largest creditor with the largest debtor and transfer the minimum of their absolute balances until all balances are zero. 7. Pseudocode explanation with math (each transfer t from debtor d to creditor c reduces balances by $t$): $$t=\min(-B_d,B_c)$$ Then update $$B_d\leftarrow B_d + t$$ $$B_c\leftarrow B_c - t$$ Repeat until all $B_i=0$. 8. Example setup: three people Roz (R), Kid A (K1), Kid B (K2). Transactions: K1 paid 120 for Roz, K2 paid 80 for Roz, and K1 paid 30 for K2. 9. Build $M$ entries: $M_{K1,R}=120$, $M_{K2,R}=80$, $M_{K1,K2}=30$, all other $M_{i,j}=0$. 10. Compute balances: $B_{K1}=\sum_j M_{K1,j}-\sum_j M_{j,K1}=120+30-0=150$ $B_{K2}=80-30=50$ $B_{R}=0-(120+80)=-200$ 11. Interpretation: K1 should receive 150, K2 should receive 50, Roz owes total 200 and pays K1 150 and K2 50. 12. To compute for your real case, please supply the list of transactions in the form (payer, beneficiary, amount) and I will compute the exact settlement for everyone.