2012 AMC 10A Problem 12

Question

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?

Friday

Saturday

Sunday

Monday

Tuesday

Count day shifts modulo

7

In

200

years, include leap days but exclude

1900

On a typical year with

365

days, moving one year into the future moves the day of the week forward one since

365 = 52 \cdot 7 + 1.

On a leap year however, the day of the week gets moved forward twice since there is an extra day.

Fifty of the years between

2012

and

1812

are multiples of

4,

but we have to discard

1900,

which leaves us with

49

leap years.

Therefore, moving back

200

years means we have to go back

200 + 49 = 249 = 35 \cdot 7 + 4

days, which means we move back

4

days in the week.

This takes us back to Friday, which is the day that corresponds to February

7,

1812.

Thus, A is the correct answer.

Problem 12 in other years