Prove that any date in March of a year falls on the same day of the week as…

Concept: Two dates fall on the same day of the week exactly when the number of days between them is a whole multiple of 7 — that is, when the “odd days” (the remainder left after dividing the elapsed days by 7) equal zero. A remainder of 0 means the weekday repeats; any other remainder shifts the weekday forward by that many days.

Application: Count the days lying between 1st March and 1st November of the same year. This span covers eight complete months — March, April, May, June, July, August, September, and October. Because the span starts on 1st March, February never falls inside it, so the count does not depend on whether the year is a leap year.

1. Add the lengths of these eight months: 31 (Mar) + 30 (Apr) + 31 (May) + 30 (Jun) + 31 (Jul) + 31 (Aug) + 30 (Sep) + 31 (Oct).

2. This totals 245 days.

3. Divide by 7 to find the odd days: 245 ÷ 7 = 35 exactly, remainder 0.

4. Zero odd days means the weekday repeats exactly after this span, so 1st November falls on the same day of the week as 1st March — and, by the identical reasoning, every other date in March coincides with its corresponding date in November of the same year.

Cross-check: Take a concrete year to confirm the result. 1st March 2024 fell on a Friday, and 245 days later — exactly 35 complete weeks — is 1st November 2024, which also fell on a Friday, matching the conclusion above.

Note: this correspondence holds for dates 1st–30th. November has no 31st, so 31st March has no matching date in November to compare against.