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 because in Brazil, the date 08/10 means October 8th, and in England, it means August 10th. 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 25th.

(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 do the longest periods occur during which the two friends cannot write to each other?
(a) (i) 03/12 means March 12th for Ralph and December 3rd for Jorge, therefore, it is an ambiguous date.
(ii) 18/08 can only be August 18th.

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

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

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

- in April, they cannot write to each other from April 1 to April 12, except for April 4;
- in September, they cannot write to each other from September 1 to September 12, except for September 9.