Step 1: Understand when the Sun is over the equator. The Sun's vertical rays hit the equator only on the two equinox days, when day and night are equal all over the world.
Step 2: Recall the cause. Because the Earth's axis is tilted, the overhead Sun shifts between the two tropics through the year and crosses the equator twice.
Step 3: Name the two crossings. One crossing is the spring (vernal) equinox and the other is the autumn equinox.
Step 4: Fix the dates. The spring equinox falls about 21 March and the autumn equinox about 23 September.
Step 5: Match with the options. Only one choice gives 21 March and 23 September.
Step 6: Conclude. These are the days the direct rays of the Sun fall on the equator. \[ \boxed{21^{st}\ \text{March and } 23^{rd}\ \text{September}} \]