The days of the week (d.o.w) which are Sunday, Monday, Tuesday,Wednesday,Thursday,Friday and Saturday (in that order) are respectively denoted by the numbers 0,1,2,3,4,5 and 6 . Any given year commencing with a particular d.o.w is assigned that value corresponding to that d.o.w. For example, the value ‘0’ would be assigned to a year commencing with a Sunday.

# A year is defined as ‘Matched’ if the remainder obtained, when the year is divided by 7, corresponds precisely with the value assigned to that particular year. For example, 2003 A.D. is NOT A Matched year since 2003 leaves a remainder of 1 upon division by 7 but January 1,2003 occurred on a Wednesday which is denoted by 3.

Determine the total number of ‘Matched’ years between 1960 A.D. and 2560 A.D.(both years inclusive) in accordance with the Western (Gregorian) Calendar System.

(In reply to 1776 by Justin)

A 28-year cycle has 4 matching years.  A period of 87 years has three such cycles plus 3 more years.  1776 had a match only if you consider the date according to the Julian calendar in effect at that time.  But then so did 1773 through 1775.  A period of 87 years starting in 1773 would have 3*4+3 = 15 matching years. 

In the present era, any period of 87 years that's uninterupted by a non-leap year divisible by 4 that starts at the first or second of four consecutive matching years will have 15 matching years, such as the following, using Tristan's x-values, where 0 indicates a matching year:

1994 0
1995 0
1996 0
1997 6
1998 6
1999 6
2000 6
2001 5
2002 5
2003 5
2004 5
2005 4
2006 4
2007 4
2008 4
2009 3
2010 3
2011 3
2012 3
2013 2
2014 2
2015 2

2016 2
2017 1
2018 1
2019 1
2020 1
2021 0
2022 0
2023 0
2024 0
2025 6
2026 6
2027 6
2028 6
2029 5
2030 5
2031 5
2032 5
2033 4
2034 4
2035 4
2036 4
2037 3
2038 3
2039 3
2040 3
2041 2
2042 2
2043 2

2044 2
2045 1
2046 1
2047 1
2048 1
2049 0
2050 0
2051 0
2052 0
2053 6
2054 6
2055 6
2056 6
2057 5
2058 5
2059 5
2060 5
2061 4
2062 4
2063 4
2064 4
2065 3
2066 3
2067 3
2068 3
2069 2
2070 2
2071 2

2072 2
2073 1
2074 1
2075 1
2076 1
2077 0
2078 0
2079 0
2080 0

In fact, starting in 2518, due to the Gregorian correction at 2600, there will be 16 matching years in the 87 year period:

2518 6
2519 6

2520 6
2521 5
2522 5
2523 5
2524 5
2525 4
2526 4
2527 4
2528 4
2529 3
2530 3
2531 3
2532 3
2533 2
2534 2
2535 2
2536 2
2537 1
2538 1
2539 1
2540 1
2541 0
2542 0
2543 0
2544 0
2545 6
2546 6
2547 6

2548 6
2549 5
2550 5
2551 5
2552 5
2553 4
2554 4
2555 4
2556 4
2557 3
2558 3
2559 3
2560 3
2561 2
2562 2
2563 2
2564 2
2565 1
2566 1
2567 1
2568 1
2569 0
2570 0
2571 0
2572 0
2573 6
2574 6
2575 6

2576 6
2577 5
2578 5
2579 5
2580 5
2581 4
2582 4
2583 4
2584 4
2585 3
2586 3
2587 3
2588 3
2589 2
2590 2
2591 2
2592 2
2593 1
2594 1
2595 1
2596 1
2597 0
2598 0
2599 0
2600 0
2601 0
2602 0
2603 0

2604 0

Posted by Charlie on 2006-01-22 17:54:49