CONO-SUR 2004 Problem 4

Arnaldo selects a nonnegative integer $a$ and Bernaldo selects a nonnegative integer $b$ . Both of them secretly tell their number to Cernaldo, who writes the numbers $5$ , $8$ , and $15$ on the board, one of them being the sum $a+b$ . Cernaldo rings a bell and Arnaldo and Bernaldo, individually, write on different slips of paper whether they know or not which of the numbers on the board is the sum $a+b$ and they turn them in to Cernaldo. If both of the papers say NO, Cernaldo rings the bell again and the process is repeated. It is known that both Arnaldo and Bernaldo are honest and intelligent. What is the maximum number of times that the bell can be rung until one of them knows the sum?

Personal note: They really phoned it in with the names there…