To determine the day of the week for a future date, one can use the concept of calculating the number of odd days. If January 1, 2001, was a Monday, then subsequent years add odd days based on whether they are leap years or not. By calculating the total number of odd days from a reference date to June 7, 2099, and finding the remainder when divided by 7, the day of the week can be ascertained. June 7, 2099, turns out to be a Sunday.