AMC 10 · 2006 · #16
Grade 4 arithmeticPick an answer.
AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Measure the sixteen-year gap
From 2004 to 2020 is 16 years, so I need the total days between the two Leap Days, then its remainder after dividing by 7.
The weekday only cares about the leftover days after packing in as many full 7-day weeks as possible.
4.OA.A.3Identify SubproblemsEach year pushes the weekday forward one day
A common year has 365 = 52 × 7 + 1 days, so each year nudges the weekday forward 1 — that is 16 extra days over the 16 years.
A year is one day more than a whole number of weeks, so the calendar slips forward by one weekday each year.
4.NBT.B.6Look For A PatternAdd the bonus day from each Leap Day
Every February 29 crossed adds one more day, and the Leap Days 2008, 2012, 2016 and 2020 make 4 extra days.
Every February 29 you cross is one bonus day the calendar has to shift by.
4.OA.A.3Identify SubproblemsTotal the shift and take the remainder
The total shift is 16 + 4 = 20 days, and 20 = 2 × 7 + 6, so the two full weeks drop out and only 6 days move the weekday.
Throwing away whole weeks costs nothing, so only the remainder after dividing by 7 matters.
Throwing away whole weeks costs nothing, so only the remainder after dividing by seven matters.
▸ Why?
After seven days everything returns to the same weekday, so full weeks leave no trace.
▸ Why?
What is left over after removing those full cycles is exactly the remainder.
Count six days forward from Sunday
Six days on from Sunday: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday — choice (E).
Six steps around the 7-day cycle from Sunday lands on the day just before Sunday, namely Saturday.
4.OA.C.5Eliminate PossibilitiesTo 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.
- Measure the sixteen-year gap
- Each year pushes the weekday forward one day
- Add the bonus day from each Leap Day
- Total the shift and take the remainder
- Count six days forward from Sunday