The Seven Bridges of Königsberg/Section 14
My procedure for determining whether in any given system of rivers and bridges it is possible to cross each bridge exactly once is as follows: 1. First I designate the individual regions separated from one another by the water as A, B, C, etc. 2. I take the total number of bridges, increase it by one, and write the resulting number uppermost. 3. Under this number I write the letters A, B, C, etc., and opposite each of these I note the number of bridges that lead to that particular region. 4. I place an asterisk next the letters that have even numbers opposite them. 5. Opposite each even number I write the half of that number and opposite each odd number I write half of the sum formed by that number plus one. 6. I add up the last column of numbers. If the sum is one less than, or equal to the number written at the top, I conclude that the required journey can be made. But it must be noted that when the sum is one less than the number at the top, the route must start from a region marked with an asterisk. And in the other case, when these two numbers are equal, it must start from a region that does not have an asterisk.
For the Königsberg problem I would set up the tabulation as follows:
| Number of bridges 7, giving 8 (= 7 + 1 ) bridges | ||
| A, | 5 | 3 |
| B, | 3 | 2 |
| C, | 3 | 2 |
| D, | 3 | 2 |
The last column now adds up to more than 8, and hence the required journey cannot be made.