AMC 10 · 2010 · #10
Grade 4 arithmeticPick an answer.
AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #5 (Look for a Pattern): each year the same date marches forward a fixed number of weekdays, so once you know the yearly step you can just walk the days Tue, Wed, Thu, ... until you hit Saturday. Tool #7 (Subproblems): first figure out the shift for a common year, then the shift for the leap-year case — two small facts that build the whole table. Tool #3 (Eliminate): the target Saturday can be skipped over in a leap-year jump, so checking each year one by one rules out the choices where May 27 is not yet a Saturday.
Common year shifts weekday by 1
A common year is 365 = 52 × 7 + 1 days — 52 whole weeks plus 1 day, so the same date lands one weekday later.
Whole weeks bring you back to the same weekday; only the leftover day past those weeks moves the date.
Whole weeks bring you back to the same weekday, so only the leftover days move the date.
▸ Why?
After seven days everything returns to where it began, so full weeks leave no trace.
▸ Why?
What is left after removing those full weeks is exactly the remainder.
Leap year shifts weekday by 2
A leap year is 366 = 52 × 7 + 2 days, and Feb 29 sits before May 27, so stepping into 2012 or 2016 shifts +2.
An extra leap day inside the year pushes the date forward one more weekday than usual.
4.OA.B.4Identify SubproblemsStep forward year by year
From Tuesday 2008, add +1 a year and +2 into 2012 and 2016: Wed, Thu, Fri, Sun, Mon, Tue, Wed, Fri, Sat.
Following one simple step-rule year after year turns the calendar into a countable chain of weekdays.
4.OA.C.5Look For A PatternRead off the first Saturday
Saturday was leaped over by the 2012 jump (Fri straight to Sun), so it first returns in 2017 — choice (E).
Because a leap-year jump can hop over a weekday, the first Saturday can land later than a naive count would guess.
4.OA.C.5Eliminate PossibilitiesA date slides forward one weekday each year, or two after a leap year — so just walk the weekdays year by year until you reach the one you want.
- Common year shifts weekday by 1
- Leap year shifts weekday by 2
- Step forward year by year
- Read off the first Saturday