Jorge, who lives in Recife, corresponds with his English friend Ralph who lives in England. The two understand each other very well in both languages, but they have a problem with dates: the date $08 / 10$ in Brazil means October 8, and in England August 10. Because of this, the two agreed not to write to each other on days when the date is ambiguous. They prefer dates like $25 / 03$ which can only mean March 25.

(a) On which of the following dates can Jorge and Ralph not write to each other?
(i) December 3
(ii) August 18
(iii) May 5

(b) When does the longest period occur during which the two friends cannot write to each other?

## List 7
(a) (i) $03 / 12$ means March 12th for Ralph and December 3rd for Jorge; thus, it is ambiguous.

(ii) 18/08 can only be August 18th.

(iii) $05 / 05$ can only be May 5th.

Therefore, (i) is a date they cannot write to each other.

(b) A date is only ambiguous when the day number can also represent a month number, which is when it is a number from 1 to 12. On the other hand, there is no ambiguity in these numbers when the month number is the same as the day number, for example, 05/05, which can only be May 5th. Therefore, in each month, they need to avoid 11 days. Thus, the longest periods they cannot write to each other occur over 11 consecutive days in January from January 2nd to January 12th, and in December, from December 2nd to December 12th. Note that in other months, the periods they cannot write to each other are shorter, see the examples:

- in April, they cannot write to each other from April 1st to April 3rd and then from April 5th to April 12th.
- in September, they cannot write to each other from September 1st to September 8th and then from September 10th to September 12th.