The Seven Bridges of Königsberg/Section 13

Every route must, of course, start in some one region, thus from the number of bridges that lead to each region I determine the number of times that the corresponding letter will occur in the expression for the entire route as follows: When the number of the bridges is odd I increase it by one and divide by two; when the number is even I simply divide it by two. Then if the sum of the resulting numbers is equal to the actual number of bridges plus one, the journey can be accomplished, though it must start in a region approached by an odd number of bridges. But if the sum is one less than the number of bridges plus one, the journey is feasible if its starting point is a region approached by an even number of bridges, for in that case the sum is again increased by one.