Combinatorics · Invariant · Parity · Divisibility · Exchange process
Peter keeps \(111\) red and \(111\) blue marbles at home; they are made by his uncle. Every day Peter may visit his uncle and carry out one exchange: either he hands over \(11\) red marbles and receives \(7\) blue ones, or he hands over \(20\) blue marbles and receives \(28\) red ones.
(a) Can Peter, after some number of days, have increased his total number of marbles by \(20\)?
(b) Can he have increased the total number of marbles by \(33\)?
(c) Can he reach a moment at which he has three times as many blue marbles as red ones?
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), regijsko (regional) round 2011, 1. letnik, category A, problem B3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source