A "coincidental day" corresponds to a date when both the date and its reversal, expressed in the mm-dd-yyyy format, shares the same day of the week.

For example, 09-22-2020 (in mm-dd-yyyy format) will occur on a Tuesday. Reversing the digits and expressing it in mm-dd-yyyy format, we obtain 02-02-2290 which will occur on a Sunday. Therefore, September 22, 2020 is NOT a coincidental day.

(i) Determine the total number of coincidental days between January 1, 2001 and December 31, 2099.

(ii) Determine the number of coincidental days that fall on a Sunday in the period covered under (i).

(iii) What is the last coincidental day in the period covered under (i)?

(i) Requiring both the date and its reversal to be between January 1, 2001 and December 31, 2099, there is a total of 24 "coincidental days".

Excluding days where the date and its reversal are the same date, there are 12 (six pairs of) "coincedental days":

[ 1] 02/02/2001 -  Friday  - 10/02/2020 [ 7]
[ 2] 06/02/2001 - Saturday - 10/02/2060 [ 8]
[ 3] 05/02/2010 -  Sunday  - 01/02/2050 [ 9]
[ 4] 04/02/2020 - Thursday - 02/02/2040 [10]
[ 5] 08/02/2021 -  Monday  - 12/02/2080 [11]
[ 6] 09/02/2040 -  Sunday  - 04/02/2090 [12]

The twelve "palindromic coincidental dates" (where the date and its reversal are the same date) are:

[ 1] 01/02/2010 - Saturday 
[ 2] 02/02/2020 -  Sunday  
[ 3] 03/02/2030 - Saturday 
[ 4] 04/02/2040 -  Monday  
[ 5] 05/02/2050 -  Monday  
[ 6] 06/02/2060 - Wednesday
[ 7] 07/02/2070 - Wednesday
[ 8] 08/02/2080 -  Friday  
[ 9] 09/02/2090 - Saturday 
[10] 10/02/2001 -  Tuesday 
[11] 11/02/2011 - Wednesday
[12] 12/02/2021 - Thursday 


(ii) Requiring both the date and its reversal to be between January 1, 2001 and December 31, 2099, there is a total of 5 "coincidental day" Sundays:
02/02/2020     
05/02/2010 <-> 01/02/2050
09/02/2040 <-> 04/02/2090


(iii) Requiring both the date and its reversal to be between January 1, 2001 and December 31, 2099 and excluding days where the date and its reversal are the same date, the last "coincidental day" is Sunday, 04/02/2090. 
Including the dates where their date and their reversal are the same, the last "coincidental date" is Saturday, 09/02/2090.

Posted by Dej Mar on 2012-02-19 15:06:03