AMC 10 · 2012 · #9
Grade 4 number-theoryPick an answer.
The end state is known (Tuesday, February 7, 2012) and we want the start state 200 years earlier, so Work Backwards fits exactly. The engine is a weekly pattern in how a fixed date shifts weekday from year to year; counting how many of those 200 years are leap years is a clean subproblem, and the answer is one of five listed days.
Pin down the birth year
Subtracting gives the birth year.
Going back 200 anniversaries just means subtracting 200 from the year while keeping February 7.
4.OA.A.3Work BackwardsHow a date shifts weekday each year
An ordinary year pushes the weekday by one.
Whole weeks return you to the same weekday, so only the remainder after dividing by 7 moves the day.
Whole weeks return you to the same weekday, so only the leftover after dividing by seven moves the day.
▸ Why?
The weekdays repeat exactly every seven days, so a full week changes nothing.
▸ Why?
Any number of days splits into full weeks plus one remainder, and only that remainder counts.
Count the leap years crossed
Counting the leap years takes the century rule.
Only leap days add the extra shift, and the divisible-by-100 rule quietly removes 1900.
4.OA.B.4Identify SubproblemsAdd the shifts and step back
Stepping back gives Friday.
The birthday is 4 weekdays before 2012's Tuesday, so count 4 days backward to reach it.
4.OA.A.3Work BackwardsTo find a weekday far in the past, count how many days the date drifts each year, keep only the leftover after dividing by 7, and step that many days backward.
- Pin down the birth year
- How a date shifts weekday each year
- Count the leap years crossed
- Add the shifts and step back