A Story of Colored Polygons and Arithmetic Progressions

I recall April of 1970. The 30 jury members of the Fourth Soviet Union National Mathematical Olympiad, of which I was one, stayed at a fabulous white castle with a white watch tower, halfway between the cities of Simferopol and Alushta, nestled in the sunny hills of Crimea, surrounded by the Black Sea. This castle should be familiar to movie buffs: in 1934, the Russian classic film Vesyolye Rebyata (Jolly Fellows) was filmed here by Sergei Eisenstein’s long-term assistant, director Grigori Aleksandrov. The problems had been selected and sent to printers. The Olympiad was to take place a day later when something shocking occurred.