100. French cities. Dr. Sharadek, who knew strategy well, was interested in the latest war and in 1940 became acquainted with the map of the French theater of military operations. From this, the following problem probably arose. The distance (as the crow flies, as are all distances in this problem) from Chalon to Vitry is 30 km, from Vitry to Chaumont 80 km, from Chaumont to Saint-Quentin 236 km, from Saint-Quentin to Reims 86 km, from Reims to Chalon 40 km. Calculate in this closed polygon the distance from Reims to Chaumont. Only Silvester Sharadek can do this without a map!
100. The five cities of Chalon, Vitry, Chaumont, Sän-Cantän, and Reims form a closed pentagon; one of its sides (the side Chaumont - Sän-Cantän) is the sum of the other four, since $236=86+40+30+80$. This is only possible when the vertices of the pentagon lie on a straight line. The cities are arranged on the line in the following order: Sän-Cantän, Reims, Chalon, Vitry, and Chaumont. The distance between Reims and Chaumont, therefore, is $40+30+80=150$ km.