It is observed that January 1, 2002 was the first "sum-day" of the 21st century, that is, when that date is written in the mm-dd-yy format, it becomes 01/01/02, and 1+1 = 2. More specifically, a "sum-day" is a date in which the month and the day add up to the year.

(i) Determine the total number of sum-days from January 1, 2002 to December 31, 2100 inclusively.

(ii) Determine the last sum-day in the period covered under (i).

The total number of sum-days, as defined, is equal the number of days in a non-leap year: 365.

The last sum-day will be December 31, 2043
(12/31/43 :: 12 + 31 = 43). In the Gregorian calendar, December is represented by the maximum value of a month MM [12] and also has the maximum number of days in a month DD [31]. Thus, is the largest sum possible of MM + DD for YY is 43. And, therefore, no later year in the century would have a sum-day.)

The 365 sum-day dates are:
01/01/02, 01/02/03, 02/01/03, 01/03/04, 02/02/04, 03/01/04, 01/04/05, 02/03/05, 03/02/05, 04/01/05, 01/05/06, 02/04/06, 03/03/06, 04/02/06, 05/01/06, 01/06/07, 02/05/07, 03/04/07, 04/03/07, 05/02/07, 06/01/07, 01/07/08, 02/06/08, 03/05/08, 04/04/08, 05/03/08, 06/02/08, 07/01/08, 01/08/09, 02/07/09, 03/06/09, 04/05/09, 05/04/09, 06/03/09, 07/02/09, 08/01/09, 01/09/10, 02/08/10, 03/07/10, 04/06/10, 05/05/10, 06/04/10, 07/03/10, 08/02/10, 09/01/10, 01/10/11, 02/09/11, 03/08/11, 04/07/11, 05/06/11, 06/05/11, 07/04/11, 08/03/11, 09/02/11, 10/01/11, 01/11/12, 02/10/12, 03/09/12, 04/08/12, 05/07/12, 06/06/12, 07/05/12, 08/04/12, 09/03/12, 10/02/12, 11/01/12, 01/12/13, 02/11/13, 03/10/13, 04/09/13, 05/08/13, 06/07/13, 07/06/13, 08/05/13, 09/04/13, 10/03/13, 11/02/13, 12/01/13, 01/13/14, 02/12/14, 03/11/14, 04/10/14, 05/09/14, 06/08/14, 07/07/14, 08/06/14, 09/05/14, 10/04/14, 11/03/14, 12/02/14, 01/14/15, 02/13/15, 03/12/15, 04/11/15, 05/10/15, 06/09/15, 07/08/15, 08/07/15, 09/06/15, 10/05/15, 11/04/15, 12/03/15, 01/15/16, 02/14/16, 03/13/16, 04/12/16, 05/11/16, 06/10/16, 07/09/16, 08/08/16, 09/07/16, 10/06/16, 11/05/16, 12/04/16, 01/16/17, 02/15/17, 03/14/17, 04/13/17, 05/12/17, 06/11/17, 07/10/17, 08/09/17, 09/08/17, 10/07/17, 11/06/17, 12/05/17, 01/17/18, 02/16/18, 03/15/18, 04/14/18, 05/13/18, 06/12/18, 07/11/18, 08/10/18, 09/09/18, 10/08/18, 11/07/18, 12/06/18, 01/18/19, 02/17/19, 03/16/19, 04/15/19, 05/14/19, 06/13/19, 07/12/19, 08/11/19, 09/10/19, 10/09/19, 11/08/19, 12/07/19, 01/19/20, 02/18/20, 03/17/20, 04/16/20, 05/15/20, 06/14/20, 07/13/20, 08/12/20, 09/11/20, 10/10/20, 11/09/20, 12/08/20, 01/20/21, 02/19/21, 03/18/21, 04/17/21, 05/16/21, 06/15/21, 07/14/21, 08/13/21, 09/12/21, 10/11/21, 11/10/21, 12/09/21, 01/21/22, 02/20/22, 03/19/22, 04/18/22, 05/17/22, 06/16/22, 07/15/22, 08/14/22, 09/13/22, 10/12/22, 11/11/22, 12/10/22, 01/22/23, 02/21/23, 03/20/23, 04/19/23, 05/18/23, 06/17/23, 07/16/23, 08/15/23, 09/14/23, 10/13/23, 11/12/23, 12/11/23, 01/23/24, 02/22/24, 03/21/24, 04/20/24, 05/19/24, 06/18/24, 07/17/24, 08/16/24, 09/15/24, 10/14/24, 11/13/24, 12/12/24, 01/24/25, 02/23/25, 03/22/25, 04/21/25, 05/20/25, 06/19/25, 07/18/25, 08/17/25, 09/16/25, 10/15/25, 11/14/25, 12/13/25, 01/25/26, 02/24/26, 03/23/26, 04/22/26, 05/21/26, 06/20/26, 07/19/26, 08/18/26, 09/17/26, 10/16/26, 11/15/26, 12/14/26, 01/26/27, 02/25/27, 03/24/27, 04/23/27, 05/22/27, 06/21/27, 07/20/27, 08/19/27, 09/18/27, 10/17/27, 11/16/27, 12/15/27, 01/27/28, 02/26/28, 03/25/28, 04/24/28, 05/23/28, 06/22/28, 07/21/28, 08/20/28, 09/19/28, 10/18/28, 11/17/28, 12/16/28, 01/28/29, 02/27/29, 03/26/29, 04/25/29, 05/24/29, 06/23/29, 07/22/29, 08/21/29, 09/20/29, 10/19/29, 11/18/29, 12/17/29, 01/29/30, 02/28/30, 03/27/30, 04/26/30, 05/25/30, 06/24/30, 07/23/30, 08/22/30, 09/21/30, 10/20/30, 11/19/30, 12/18/30, 01/30/31, 03/28/31, 04/27/31, 05/26/31, 06/25/31, 07/24/31, 08/23/31, 09/22/31, 10/21/31, 11/20/31, 12/19/31, 01/31/32, 03/29/32, 04/28/32, 05/27/32, 06/26/32, 07/25/32, 08/24/32, 09/23/32, 10/22/32, 11/21/32, 12/20/32, 03/30/33, 04/29/33, 05/28/33, 06/27/33, 07/26/33, 08/25/33, 09/24/33, 10/23/33, 11/22/33, 12/21/33, 03/31/34, 04/30/34, 05/29/34, 06/28/34, 07/27/34, 08/26/34, 09/25/34, 10/24/34, 11/23/34, 12/22/34, 05/30/35, 06/29/35, 07/28/35, 08/27/35, 09/26/35, 10/25/35, 11/24/35, 12/23/35, 05/31/36, 06/30/36, 07/29/36, 08/28/36, 09/27/36, 10/26/36, 11/25/36, 12/24/36, 07/30/37, 08/29/37, 09/28/37, 10/27/37, 11/26/37, 12/25/37, 07/31/38, 08/30/38, 09/29/38, 10/28/38, 11/27/38, 12/26/38, 08/31/39, 09/30/39, 10/29/39, 11/28/39, 12/27/39, 10/30/40, 11/29/40, 12/28/40, 10/31/41, 11/30/41, 12/29/41, 12/30/42, 12/31/43

Posted by Dej Mar on 2010-01-14 21:28:18