### The "Every Other Month" Deal

How many times has this happened to you?

You and a roommate or lover choose each to pay a monthly payment every other month. That is, you trade off payments. One month you pay, and the next month, your friend or lover pays.

And then, eventually, you break up, and you've forgotten who made the first payment, or whether an even or odd number of payments have been made.

With this kind of arrangement, you two are only "even" after every *even* payment. The 2nd, the 4th, the 6th, etc.

If an odd number of payments are actually made, then whoever makes the last payment is actually owed half of it by the other party. But who remembers? And, what about all the interest lost by the party who makes the odd payments, given the "even" party is effectively borrowing money from them half of the time?

You can avoid this, of course, by splitting all the payments in half such that both parties pay. But that means twice as many checks, or withdrawals, or transactions. The whole point is to avoid that.

Consider this solution. Say the monthly payment is $100.

First party (odd party) pays the first payment, and is thereby owed $50 by the even party. But now, the even party pays $25 to the odd party.

So the party who made the last payment is owed a quarter payment to make things even.

Now, the second party (even party) pays the second payment. Since he's paying half of it for the sake of the odd party, he pays off his $25 debt, and effectively lends $25 to the odd party.

So, the invariant condition is that whoever made the last payment is owed 1/4 payment by the other party. To settle up, the other party just pays off this loan.

In summary: Whoever makes the first payment gets paid 1/4 payment by the other party, and whoever makes the last payment gets paid 1/4 payment by the other party.

And there's no need to remember who paid first. Just remember you are doing the "quarter payment" deal.

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