1. Phase transition in the Countdown problem
Lucas Lacasa, School of Mathematical Sciences, Queen Mary University of London Bartolo Luque, Department of Applied Mathematics, Technical University of Madrid
RULES
Recursive algorithm
- Given a set of k integers extracted randomly from the pool [1,M]
- Extract another random integer T from the pool [1,M]
- Combine the set of integers through arithmetic operations (use at most once each number)
Question WHICH IS THE PROBABILITY OF REACHING T ?
Probability of winning the game
(Rescaled) intensive control parameter
Example [k=4,M=200]
{3, 12, 5, 53} T=118
Solution exists? YES S=(53-12)*3-5=118
{7, 11, 82, 26} T=181
Solution exists? NO
(Closest formula= 26*7=182)
Efficiency is maximal at the transition
The transition between loosing and winning is sharp (zero-one law)
The problem is similar to optimization problems (random K-satisfiability)
Efficiency is maximal close to the transition point
Connections between statistical physics of spin glasses, number theory and combinatorial group theory
Null model for cognitive processes (arithmetics)
In the finite size system analogous to the quiz show, the threshold between impossible (P=0) and easy (P=1) is for k=6 numbers!!! (see figure on the top)
Q denotes the average amount of accessible targets in [1,M] per integer
Lucas Lacasa and Bartolo Luque, Phase transition in the Countdown problem, Physical Review E 86, (R) (2012)