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2012 AMC 10A Problem 12

2012 AMC 10A Problem 12AMC 10 Step by Step

Name: Date: September 13, 2026

Clocks, Calendars & Time · difficulty 3 of 5 · about 4 min · position P11-15

A year is a leap year if and only if the year number is divisible by $400$ (such as $2000$ ) or is divisible by $4$ but not $100$ (such as $2012$ ). The $200$ th anniversary of the birth of novelist Charles Dickens was celebrated on February $7$ , $2012$ , a Tuesday. On what day of the week was Dickens born?

A)
$\text{Friday}$
B)
$\text{Saturday}$
C)
$\text{Sunday}$
D)
$\text{Monday}$
E)
$\text{Tuesday}$

Answer

Key insight

Every year shifts the weekday by one, plus one per leap day; 200 years and 49 leap days give 249, which is 4 mod 7: back 4 from Tuesday.

Solution

Dickens was born on February $7$ , $1812$ . Count the days from that date to February $7$ , $2012$ .

A $365$ -day year is $52$ weeks plus $1$ day, so each year advances the weekday by $1$ , and every February $29$ in the interval adds one extra day. Over $200$ years the base shift is $200$ days.

Leap days between the two dates are the February $29$ 's of years $1812$ through $2008$ (Feb 29, 1812 comes after Feb 7, 1812, so it counts; Feb 29, 2012 comes after Feb 7, 2012, so it does not). Multiples of $4$ from $1812$ to $2008$ : $\frac{2008 - 1812}{4} + 1 = 50$ . The year $1900$ is divisible by $100$ but not $400$ , so it is not a leap year; that leaves $49$ leap days.

Total shift: $200 + 49 = 249$ days, and $249 = 7 \cdot 35 + 4$ , so the anniversary is $4$ weekdays later in the week than the birthday. Going back $4$ days from Tuesday: Monday, Sunday, Saturday, Friday.

The answer is $\boxed{\textbf{(A)}\ \text{Friday}}$ .

Why this works

Weekday problems reduce to counting days mod $7$ , and since $365 \equiv 1$ , only the number of years and the number of leap days matter. The problem restates the leap-year rule precisely so that you notice the $1900$ exception. Decide carefully which leap days fall strictly inside the interval, then move in the correct direction (backwards to an earlier date).

The trap

Counting 50 leap years by forgetting that 1900 is not a leap year, or counting Feb 29, 1812 (which falls after the birthday) incorrectly.

Common mistakes

Counting 50 leap years by forgetting that 1900 is not a leap year, or counting Feb 29, 1812 (which falls after the birthday) incorrectly.
Moving forward 4 days from Tuesday to Saturday instead of backwards to Friday.

Techniques

Set up the equation/formula and compute; no special trick needed