Henry Ernest Dudeney/Puzzles and Curious Problems/346 - A Leap Year Puzzle/Solution
Puzzles and Curious Problems by Henry Ernest Dudeney: $346$
- A Leap Year Puzzle
- The month of February in $1928$ contained five Wednesdays.
- There is, of course, nothing remarkable in this fact, but it will be found interesting to discover
- when was the last year and when will be the next year that had, and that will have, $5$ Wednesdays in February.
Solution
The next year after $1928$ that contains $5$ February Wednesdays was $1956$.
The one immediately before $1928$ was $1888$.
Proof
$5$ February Wednesdays can occur only in a leap year.
A leap year February contains $29$ days.
Hence such a February always has exactly one day of the week of which there are $5$.
This happens on $1$st, $8$th, $15$th, $22$nd and $29$th February.
So the puzzle is equivalent to determining on which years $29$th February falls on a Wednesday.
Except when the year is divisible by $100$ but not $400$, leap years occur every $4$ years.
Each consecutive set of $4$ years in the Gregorian calendar contains $3 \times 365 + 366 = 1461$ days.
This is congruent to $5$ modulo $7$.
So, on consecutive leap years, $29$th February happens on:
- $(1): \quad$ Wednesday, Monday, Saturday, Thursday, Tuesday, Sunday, Friday.
Hence (apart from at the turn of the century), there are $4 \times 7 = 28$ years between consecutive years when the days coincide.
Hence after $1928$, the next one is $1956$.
$28$ years before $1928$ is $1900$.
But $1900$, as we have determined, is not a leap year.
If it were a leap year, then $28$ years before that, that is $1872$, would have had $5$ Wednesdays.
But there is a day less between $1872$ and $1900$ than there would be.
So $29$th February in $1872$ fell on a Thursday.
Hence, from the sequence of days in $(1)$ above, $29$th February falls on a Wednesday $4$ leap years later.
That is, $16$ years, which is $1888$.
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $346$. -- A Leap Year Puzzle
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $528$. A Leap Year Puzzle