A TV show airs every 28 days. This century there was a year when the show aired on both January 1 and January 29. In how many years will the show air twice in January again?
Solution
In a common year, there are days. Thus, after a common year, the dates of the airings shift 1 day earlier. After a leap year, the airings shift 2 days earlier. In every 4 years, there are 3 common years and 1 leap year, which means a total shift of days. Thus in years the total shift is days.
Thus, if in year the show airs on January 29, then in year it will air on January 4, and in the intermediate years it will air between these dates. In these years, there cannot be a second airing in January. However, in year , the first airing will be on January 2 or January 3, which also means a second airing in January. Thus 21 years is the desired answer.
Want a route through all this instead of an archive? The track
puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.