As I’m writing this, tomorrow is Chinese New Year (in the US) and today is Ash Wednesday. (I suspect it’ll be a day later when you read this.) This raises a question: does Chinese New Year often fall during Lent (that is, on or after Ash Wednesday)? The coincidence creates conflicts for many Asian Catholics: see e. g. here, here, here, here.
Chinese New Year falls on the second new moon after the winter solstice.
Ash Wednesday is 46 days before Easter. (Yes, 46. I know, you thought Lent was 40 days. Sundays don’t count.) Easter is the Sunday after the first full moon on or after the spring equinox (the “Paschal full moon.)
How far apart are these? Well, there are 90 days between the winter solstice (December 22) and the spring equinox (March 21). This is between three and three-and-a-half lunar months (of 29.5 days each), so from Chinese New Year to the Paschal full moon (i. e. the full moon on or after the spring equinox) is either one-and-a-half or two-and-a-half lunar months. In the cases when it’s one-and-a-half, Ash Wednesday will fall around Chinese New Year; when it’s two-and-a-half, Ash Wednesday will be a month or so after Chinese New Year.
The short interval happens when Chinese New Year is relatively late in the window of dates it can occur, which is January 21 to February 20. In particular, if Chinese New Year is less than about 44 days (that is, one-and-a-half lunar months) before the spring equinox (March 21), then it’s 1.5 lunar months from the Paschal full moon, and we get a situation like this year’s. That is, Chinese New Year is roughly around the beginning of Lent if it’s around February 5 or later – about half the time. Not so unusual after all.
As for the day of the week – Ash Wednesday works out to be between 39 and 45 days before the Paschal full moon. Chinese New Year is about 44 days before in half of years. So Chinese New Year can only fall within the first couple days of Lent. (If you want to do the calculations: can it fall later than Thursday? I suspect this might be possible, because the two calendars work on different rules — the Chinese calendar is based on astronomical observation whereas the Christian ecclesiastical calendar is based on computations than can be done with relatively simple arithmetic.)