A "sum digit-day" is a date in which the sum of the digits of the month and the day add up to the sum of the digits of the year in the mm-dd-yy format.

For example,12-18-84 is a sum digit-day as the respective sum of the digits of the month and the day add up to 12 which is equal to the sum of the digits of the year.

(i) Determine the total number of sum digit-days from January 1, 2002 to December 31, 2099 inclusively.

(ii) Determine the year(s) having the maximum number of sum digit- days in the period covered under (i).

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

I had done this with a computer program, but there's no point in showing that.  However it might be interesting to see the breakdown by s.o.d.  Each possible sod, from 2 to 18 is listed below, followed by the dates that satisfy that sod, and further followed by the years with the given sod.  Each list of dates and each list of years is followed by the count of the number of dates or number of years.

For example, the list shows four dates within a year where the sod is 2, and three years in which the sod is 2 for the last two digits.

 2
  1- 1  1-10 10- 1 10-10     4
2002 2011 2020      3

 3
  1- 2  1-11  1-20  2- 1  2-10 10- 2 10-11 10-20 11- 1 11-10     10
2003 2012 2021 2030      4

 4
  1- 3  1-12  1-21  1-30  2- 2  2-11  2-20  3- 1  3-10 10- 3 10-12 10-21 10-30 11- 2 11-11 11-20 12- 1 12-10     18
2004 2013 2022 2031 2040      5

 5
  1- 4  1-13  1-22  1-31  2- 3  2-12  2-21  3- 2  3-11  3-20  4- 1  4-10 10- 4 10-13 10-22 10-31 11- 3 11-12 11-21 11-30 12- 2 12-11 12-20     23
2005 2014 2023 2032 2041 2050      6

 6
  1- 5  1-14  1-23  2- 4  2-13  2-22  3- 3  3-12  3-21  3-30  4- 2  4-11  4-20  5- 1  5-10 10- 5 10-14 10-23 11- 4 11-13 11-22 12- 3 12-12 12-21 12-30     25
2006 2015 2024 2033 2042 2051 2060      7

 7
  1- 6  1-15  1-24  2- 5  2-14  2-23  3- 4  3-13  3-22  3-31  4- 3  4-12  4-21  4-30  5- 2  5-11  5-20  6- 1  6-10 10- 6 10-15 10-24 11- 5 11-14 11-23 12- 4 12-13 12-22 12-31     29
2007 2016 2025 2034 2043 2052 2061 2070      8

 8
  1- 7  1-16  1-25  2- 6  2-15  2-24  3- 5  3-14  3-23  4- 4  4-13  4-22  5- 3  5-12  5-21  5-30  6- 2  6-11  6-20  7- 1  7-10 10- 7 10-16 10-25 11- 6 11-15 11-24 12- 5 12-14 12-23     30
2008 2017 2026 2035 2044 2053 2062 2071 2080      9

 9
  1- 8  1-17  1-26  2- 7  2-16  2-25  3- 6  3-15  3-24  4- 5  4-14  4-23  5- 4  5-13  5-22  5-31  6- 3  6-12  6-21  6-30  7- 2  7-11  7-20  8- 1  8-10 10- 8 10-17 10-26 11- 7 11-16 11-25 12- 6 12-15 12-24     34
2009 2018 2027 2036 2045 2054 2063 2072 2081 2090      10

 10
  1- 9  1-18  1-27  2- 8  2-17  2-26  3- 7  3-16  3-25  4- 6  4-15  4-24  5- 5  5-14  5-23  6- 4  6-13  6-22  7- 3  7-12  7-21  7-30  8- 2  8-11  8-20  9- 1  9-10 10- 9 10-18 10-27 11- 8 11-17 11-26 12- 7 12-16 12-25     36
2019 2028 2037 2046 2055 2064 2073 2082 2091      9

 11
  1-19  1-28  2- 9  2-18  2-27  3- 8  3-17  3-26  4- 7  4-16  4-25  5- 6  5-15  5-24  6- 5  6-14  6-23  7- 4  7-13  7-22  7-31  8- 3  8-12  8-21  8-30  9- 2  9-11  9-20 10-19 10-28 11- 9 11-18 11-27 12- 8 12-17 12-26     36
2029 2038 2047 2056 2065 2074 2083 2092      8

 12
  1-29  2-19  2-28  3- 9  3-18  3-27  4- 8  4-17  4-26  5- 7  5-16  5-25  6- 6  6-15  6-24  7- 5  7-14  7-23  8- 4  8-13  8-22  8-31  9- 3  9-12  9-21  9-30 10-29 11-19 11-28 12- 9 12-18 12-27     32
2039 2048 2057 2066 2075 2084 2093      7

 13
  3-19  3-28  4- 9  4-18  4-27  5- 8  5-17  5-26  6- 7  6-16  6-25  7- 6  7-15  7-24  8- 5  8-14  8-23  9- 4  9-13  9-22 11-29 12-19 12-28     23
2049 2058 2067 2076 2085 2094      6

 14
  3-29  4-19  4-28  5- 9  5-18  5-27  6- 8  6-17  6-26  7- 7  7-16  7-25  8- 6  8-15  8-24  9- 5  9-14  9-23 12-29     19
2059 2068 2077 2086 2095      5

 15
  4-29  5-19  5-28  6- 9  6-18  6-27  7- 8  7-17  7-26  8- 7  8-16  8-25  9- 6  9-15  9-24     15
2069 2078 2087 2096      4

 16
  5-29  6-19  6-28  7- 9  7-18  7-27  8- 8  8-17  8-26  9- 7  9-16  9-25     12
2079 2088 2097      3

 17
  6-29  7-19  7-28  8- 9  8-18  8-27  9- 8  9-17  9-26     9
2089 2098      2

 18
  7-29  8-19  8-28  9- 9  9-18  9-27     6
2099      1

The products of the date counts times year counts add up to 2486, which is supplemented by 1 for 2-29-76.

Posted by Charlie on 2011-09-20 22:21:01