Amir was born on Feb. 29th of 2012 which was a Wednesday. If he lives to be…

Concept

A person born on 29 February only has a calendar birthday in a leap year, because 29 February exists only in leap years. To find which of those birthdays fall on a given weekday, track how the day of the week shifts from one 29 February to the next. The shift is governed by "odd days": the number of days beyond complete weeks in the gap between two dates determines how far the weekday advances.

Application

1. Birthdays occur only in leap years: 2016, 2020, 2024, ... — never in the three common years in between, and the birth day in 2012 is the birth, not a celebrated anniversary.

2. From one 29 February to the next (a 4-year gap with one extra leap day) there are 4 years = 4 ordinary years + 1 leap day; the total odd days are 5, so each successive leap-year birthday lands 5 weekdays later than the previous one.

3. Adding 5 odd days repeatedly cycles the weekday; the weekday pattern would repeat every 7 birthdays (5 x 7 = 35 = 0 odd days), i.e. every 28 years a 29 February returns to the same weekday — but only while the leap-year rhythm is unbroken.

4. The break: the year 2100 is divisible by 100 but not by 400, so 2100 is NOT a leap year. There is no 29 February in 2100, which disrupts the clean 28-year cycle for births later in the lifespan.

5. Working through the leap years from the first celebrated birthday up to age 101 (the lifespan reaches 2113, so leap years through 2112), the 29 February dates that fall on Wednesday are 2040, 2068, 2096 and 2108.

Cross-check

The first three Wednesdays sit 28 years apart (2040, 2068, 2096), confirming the unbroken-cycle spacing. The fourth, 2108, is only 12 years after 2096 instead of 28 — exactly because the missing 29 February in 2100 shifted the rhythm. Counting these gives four Wednesday birthdays.