algorithm - binary representation of all states in tower of hanoi -


in modified toh have 4 pegs. in turn have total of 4^n disk positions. in 1 of solutions going through, given state represented using below code - 

for(int disc = 0; disc < n; disc++){         state += tower[disc]<<(disc*2);     }

tower[disc] --> tower disc located, can (0,1,2,3)

if use n=3 in above code, gives me 63, 4^n -1. formula works, i.e 0-63 64 positions can represented. not able understand mathematical correlation.

can please explain how above formula can represent possible disk positions , how change if further change number of pegs or n lets 5.

since have 4 pegs, position of disk can encoded in 2 bits (00, 01, 10, 11 => 0, 1, 2, 3). multiplication 2 gives each disk 2 bits of independent memory space within integer state, first disk starting @ bit 0, second @ bit 2, , on... ith disk @ (i - 1) * 2. left shift << moves data each disk correct bit position.

however code can "unsafe" - if there mistake in game logic somewhere else caused value of tower greater 3, when shifted overflow designated 2 bits of space. prevent this, bitwise , clamp data [0, 3]:

state += (tower[disc] & 3) << (i * 2);

note can use bitwise or instead of addition - state |= ....

from example n = 3 provided looks plates starting peg 4 (i.e. tower[3]). if change n = 5 once again give 4^n - 1 = 1023, bits below n * 2 set 1.


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