In any calendar year there are at least four and at most seven eclipses, solar and lunar together, and they come in two or three clusters about six months apart.
The Moon’s orbit is tilted 5.1° to the Earth’s, so it crosses the plane of the Earth’s orbit at two points, and an eclipse needs a new moon (solar) or a full moon (lunar) near a crossing. The Sun passes the line of crossings every 173 days, and for about 34 days around each pass the Moon can eclipse it or the Earth’s shadow, which is enough for two or three eclipses. Two passes in a calendar year give at least four eclipses, and when one falls in early January a third pass fits in, which gives up to seven.
It counts eclipses of any kind, including a faint penumbral lunar eclipse that is hard to see. Each of the four to seven is visible from only part of the Earth: a particular place sees a total solar eclipse about once in several centuries, and an annular or partial one more often. Years with seven eclipses are rare, and some calendar years have as few as four.
Estimate with the rule, then check it against the calculator that models it properly.
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