Thanks for all the responses Buddhaboy was one of the closest to the answer. Well the number of times he has to return to the start is N = 2^(n-1) n= ((2 x Total Distance)/distance per gallon) - 1 so n = 8 Hence N = 128 Rationale: for every 25 miles he has to use half the number of gallons. At 150th mile he has to have 2 gallons of water to complete the distance. so 125 he has to have 4 gallons, similarly 100 : 8, 75 : 16, 50 : 32, 25 : 64 and thus 0 : 128.