Final answer:
An upright flagpole will cast no shadow at noon only if it is located at 23° N latitude (the Tropic of Cancer) on about June 21, which is the summer solstice.
Step-by-step explanation:
Regarding when an upright flagpole will cast no shadow because the sun is directly overhead, this phenomenon happens only at specific latitudes and particular times of the year. On or about June 21, the date known as the summer solstice, the Sun shines most directly upon the Northern Hemisphere and passes through the zenith (the point in the sky directly above) at places on Earth that are at 23° N latitude. This latitude is known as the Tropic of Cancer. Therefore, an upright flagpole would cast no shadow at noon only if located precisely at 23° N latitude on the summer solstice.
Outside the Tropic of Cancer, the sun will never be directly overhead, and thus a flagpole will always cast a shadow at noon, albeit shorter when the sun is higher in the sky, such as during the local summer months.