Problem 6
An international society has its members from six different countries. The list of members contains names, numbered . Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.
Step 4 of 6: Continue through all six countries
In plain words
The counts are chosen so that one country remains at the end.
Detailed analysis
The same argument gives successive group sizes (among differences in countries) and then (among differences in countries). At each stage, any difference in an earlier country closes the argument by unwinding the nested differences.