1. YAP: Yet Another Puzzle Boudewijn R. Haverkort November 16, 2007

2. Collatz Conjecture (1937) n = 1, 2, 3,... Sequence of n such that n := n/2 if n is even n := 3n + 1, if n is odd Conjecture: for all n ≥ 1, the above sequence returns to 1 “Always Eventually 1”

3. A few cases... Sequence of n such that n := n/2 if n is even n := 3n + 1, if n is odd 5, 16, 8, 4, 2, 1 6, 3, 10, 5, 16, 8, 4, 2, 1 27, 82, 41,...111 steps..., 4, 2, 1 Computer-checked up to 2.88 x 10 18

4. Probabilistic interpretation View sequence of n’s as DTMC on the positive natural numbers, with “probability 1” transitions pCTL-type of model checking: -- find the satisfaction set for P ≥1 {AE “1”} -- this set should be equal to { 1, 2,... } Use all we know about Markov chains, abstraction, bisimulation, infinite-state Markov chains, regular model checking,... //en.wikipedia.org/wiki/Collatz_conjecture