AMC 10 · 2006 · #16
Grade 4 arithmeticLeap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?
Pick an answer.
AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Leap Day, February 29, 2004, fell on a Sunday. Figure out which day of the week the next Leap Day sixteen years later, February 29, 2020, falls on.
Givens: February 29, 2004 was a Sunday; The target date is Leap Day, February 29, 2020; A common year has $365$ days; a leap year (a year that has a February 29) has $366$ days; Leap Days here fall on the years $2008$, $2012$, $2016$, and $2020$; Answer choices: (A) Tuesday, (B) Wednesday, (C) Thursday, (D) Friday, (E) Saturday
Unknowns: The day of the week of February 29, 2020
Understand
Restated: Leap Day, February 29, 2004, fell on a Sunday. Figure out which day of the week the next Leap Day sixteen years later, February 29, 2020, falls on.
Givens: February 29, 2004 was a Sunday; The target date is Leap Day, February 29, 2020; A common year has $365$ days; a leap year (a year that has a February 29) has $366$ days; Leap Days here fall on the years $2008$, $2012$, $2016$, and $2020$; Answer choices: (A) Tuesday, (B) Wednesday, (C) Thursday, (D) Friday, (E) Saturday
Plan
Primary tool: #5 Look for a Pattern
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
The whole problem is about how the weekday drifts forward as time passes, so Tool #5 (Look for a Pattern) is primary: the key pattern is that every ordinary year moves the weekday forward by exactly $1$ day, because $365$ is one more than a whole number of weeks. Tool #7 (Identify Subproblems) splits the total count into two easy pieces — the shift from the $16$ years and the extra shift from the leap days in between — so nothing has to be counted day by day. Tool #3 (Eliminate Possibilities) fits the multiple-choice format: once the weekday shift is known, stepping forward from Sunday points to exactly one of the five listed days.
Execute — Answer: E
4.OA.A.3 Step 1 Measure the sixteen-year gap
- From 2004 to 2020 is $2020-2004=16$ years.
- The weekday of a future date depends only on how many days have passed since a date I already know: every $7$ days the weekday cycle repeats, so what I really need is the total number of days between the two Leap Days, and then its remainder when divided by $7$.
- I will build that total in two pieces: the days coming from the whole years, and the extra days coming from the Leap Days in between.
💡 The weekday only cares about the leftover days after packing in as many full 7-day weeks as possible.
4.NBT.B.6 Step 2 Each year pushes the weekday forward one day
- A common year has $365$ days, and $365 = 52\times 7 + 1$.
- So $365$ days is exactly $52$ full weeks plus $1$ extra day, which nudges the weekday forward by $1$.
- Across the $16$ years between the two Leap Days, that is $16$ forward nudges so far, one for each year.
💡 A year is one day more than a whole number of weeks, so the calendar slips forward by one weekday each year.
4.OA.A.3 Step 3 Add the bonus day from each Leap Day
- A leap year has one extra day — the February 29 — so each Leap Day I pass adds one more forward nudge on top of the $16$.
- Between February 29, 2004 and February 29, 2020 the Leap Days occur in $2008$, $2012$, $2016$, and $2020$, which is $4$ Leap Days.
- So I add $4$ more extra days.
💡 Every February 29 you cross is one bonus day the calendar has to shift by.
4.NBT.B.6 Step 4 Total the shift and take the remainder
- The total forward shift is $16 + 4 = 20$ days.
- Since the weekday repeats every $7$ days, I divide by $7$ and keep the remainder: $20 = 2\times 7 + 6$, so the remainder is $6$.
- The two full weeks ($14$ days) change nothing; only the $6$ leftover days actually move the weekday.
💡 Throwing away whole weeks costs nothing, so only the remainder after dividing by 7 matters.
4.OA.C.5 Step 5 Count six days forward from Sunday
- Starting at Sunday and moving forward $6$ days: Monday $(1)$, Tuesday $(2)$, Wednesday $(3)$, Thursday $(4)$, Friday $(5)$, Saturday $(6)$.
- So February 29, 2020 falls on a Saturday, which is choice (E).
💡 Six steps around the 7-day cycle from Sunday lands on the day just before Sunday, namely Saturday.
4.OA.A.3 From 2004 to 2020 is $2020-2004=16$ years. The weekday of a future date depends 4.NBT.B.6 A common year has $365$ days, and $365 = 52\times 7 + 1$. So $365$ days is exact 4.OA.A.3 A leap year has one extra day — the February 29 — so each Leap Day I pass adds o 4.NBT.B.6 The total forward shift is $16 + 4 = 20$ days. Since the weekday repeats every $ 4.OA.C.5 Starting at Sunday and moving forward $6$ days: Monday $(1)$, Tuesday $(2)$, Wed Review
Reasonableness: A direct count agrees: the total number of days is $16\times 365 + 4 = 5840 + 4 = 5844$, and $5844 = 834\times 7 + 6$, again a remainder of $6$, which is Saturday. It also makes sense that the answer sits one day short of a full extra week: $20$ days is just $1$ short of $21 = 3\times 7$, so the weekday lands one day before Sunday, namely Saturday.
Alternative: Group the years into the four 4-year leap cycles $2004\to2008\to2012\to2016\to2020$. Each cycle contains $3$ common years and $1$ leap year, so it spans $3(365)+366 = 1461$ days, and $1461 = 208\times 7 + 5$, a shift of $5$ days per cycle. Four cycles give $4\times 5 = 20$ days of shift, and $20 = 2\times 7 + 6$ leaves the same remainder $6$, landing on Saturday.
CCSS standards used (min grade 4)
4.OA.A.3Solve multistep word problems with the four operations, including interpreting remainders (Setting up the 16-year gap, counting the $4$ leap days, and interpreting the final remainder as a day of the week.)4.NBT.B.6Find whole-number quotients and remainders (Seeing that $365 = 52\times 7 + 1$ so each year shifts the weekday by $1$, and reducing the total shift $20 = 2\times 7 + 6$ to a remainder of $6$.)4.OA.C.5Generate and use a number pattern that follows a given rule (Using the repeating 7-day weekday cycle to step $6$ days forward from Sunday to Saturday.)
⭐ To jump a date far into the future, count the total days — one per year plus one for every February 29 you pass — then divide by 7, because only the leftover days move the weekday forward.
⭐ To jump a date far into the future, count the total days — one per year plus one for every February 29 you pass — then divide by 7, because only the leftover days move the weekday forward.
More like this
Same archetype — closest grade level first.