The Seven Bridges of Königsberg/Section 15

by Leonhard Euler

Let us take an example of two islands, with four rivers forming the surrounding water, as shown in Figure 3. Fifteen bridges, marked a, b, c, d, etc., cross the water around the islands and the adjoining rivers; the question is whether a journey can be arranged that will pass over all the bridges, but not over any of them more than once. 1. I begin by marking all the regions that are separated from one another by the water with the letters A, B, C, D, E, F—there are six of them. 2. I take the number of bridges—15—add one and write this number—16—uppermost. 3. I write the letters A, B, C, etc. in a column and opposite each letter I

Figure 3

		16
A*,	8	4
B*,	4	2
C*,	4	2
D,	3	2
E,	5	3
F*,	6	3
		16

write the number of bridges connecting with that region, e.g., 8 bridges for A, 4 for B, etc. 4. The letters that have even numbers opposite them I mark with an asterisk. 5. In a third column I write the half of each corresponding even number, or, if the number is odd, I add one to it, and put down half the sum. 6. Finally I add the numbers in the third column and get 16 as the sum. This is the same as the number 16 that appears above, and hence it follows that the journey can be effected if it begins in regions D or E, whose symbols have no asterisk. The following expression represents such a route:

EaFbBcFdAeFfCgAhCiDkAmEnApBoElD.

Here I have also indicated, by small letters between the capitals, which bridges are crossed.