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 Analytical solution by Tristan)

Here's a table of Tristan's x-values. There are 6 columns and the year numbers are shown with 2000 subtracted, for brevity horizontally--i.e., to fit on an 80-character line.:

        7 4| 54 6|-----|148 5|195 0|241 3|289 5|335 2|382 4|430 6|477 1|523 5
-40 2|  8 4| 55 6|101 2|149 4|196 0|242 3|290 5|336 2|383 4|431 6|478 1|524 5
-39 1|  9 3| 56 6|102 2|150 4|197 6|243 3|291 5|337 1|384 4|432 6|479 1|525 4
-38 1| 10 3| 57 5|103 2|151 4|198 6|244 3|292 5|338 1|385 3|433 5|480 1|526 4
-37 1| 11 3| 58 5|104 2|152 4|199 6|245 2|293 4|339 1|386 3|434 5|481 0|527 4
-36 1| 12 3| 59 5|105 1|153 3|200 6|246 2|294 4|340 1|387 3|435 5|482 0|528 4
-35 0| 13 2| 60 5|106 1|154 3|-----|247 2|295 4|341 0|388 3|      483 0|529 3
-34 0| 14 2| 61 4|107 1|155 3|201 6|248 2|      342 0|389 2|436 5|484 0|530 3
-33 0| 15 2| 62 4|108 1|      202 6|249 1|296 4|343 0|390 2|437 4|485 6|531 3
-32 0|       63 4|109 0|156 3|203 6|250 1|297 3|344 0|391 2|438 4|486 6|532 3
-31 6| 16 2| 64 4|110 0|157 2|204 6|251 1|298 3|345 6|392 2|439 4|487 6|533 2
-30 6| 17 1| 65 3|111 0|158 2|205 5|252 1|299 3|346 6|393 1|440 4|488 6|534 2
-29 6| 18 1| 66 3|112 0|159 2|206 5|253 0|300 3|347 6|394 1|441 3|489 5|535 2
-28 6| 19 1| 67 3|113 6|160 2|207 5|254 0|-----|348 6|395 1|442 3|490 5|536 2
-27 5| 20 1| 68 3|114 6|161 1|208 5|255 0|301 3|349 5|396 1|443 3|491 5|537 1
-26 5| 21 0| 69 2|115 6|162 1|209 4|256 0|302 3|350 5|397 0|444 3|      538 1
-25 5| 22 0| 70 2|116 6|163 1|210 4|257 6|303 3|351 5|398 0|445 2|492 5|539 1
-24 5| 23 0| 71 2|117 5|164 1|211 4|258 6|304 3|      399 0|446 2|493 4|540 1
-23 4| 24 0|      118 5|165 0|      259 6|305 2|352 5|400 0|447 2|494 4|541 0
-22 4| 25 6| 72 2|119 5|166 0|212 4|260 6|306 2|353 4|401 6|448 2|495 4|542 0
-21 4| 26 6| 73 1|120 5|167 0|213 3|261 5|307 2|354 4|402 6|449 1|496 4|543 0
-20 4| 27 6| 74 1|121 4|168 0|214 3|262 5|308 2|355 4|403 6|450 1|497 3|544 0
-19 3| 28 6| 75 1|122 4|169 6|215 3|263 5|309 1|356 4|404 6|451 1|498 3|545 6
-18 3| 29 5| 76 1|123 4|170 6|216 3|264 5|310 1|357 3|405 5|452 1|499 3|546 6
-17 3| 30 5| 77 0|124 4|171 6|217 2|265 4|311 1|358 3|406 5|453 0|500 3|547 6
-16 3| 31 5| 78 0|125 3|172 6|218 2|266 4|312 1|359 3|407 5|454 0|-----|
-15 2| 32 5| 79 0|126 3|173 5|219 2|267 4|313 0|360 3|      455 0|501 3|548 6
-14 2| 33 4| 80 0|127 3|174 5|220 2|      314 0|361 2|408 5|456 0|502 3|549 5
-13 2| 34 4| 81 6|      175 5|221 1|268 4|315 0|362 2|409 4|457 6|503 3|550 5
       35 4| 82 6|128 3|176 5|222 1|269 3|316 0|363 2|410 4|458 6|504 3|551 5
-12 2| 36 4| 83 6|129 2|177 4|223 1|270 3|317 6|364 2|411 4|459 6|505 2|552 5
-11 1| 37 3| 84 6|130 2|178 4|224 1|271 3|318 6|365 1|412 4|460 6|506 2|553 4
-10 1| 38 3| 85 5|131 2|179 4|225 0|272 3|319 6|366 1|413 3|461 5|507 2|554 4
 -9 1| 39 3| 86 5|132 2|180 4|226 0|273 2|320 6|367 1|414 3|462 5|508 2|555 4
 -8 1| 40 3| 87 5|133 1|181 3|227 0|274 2|321 5|368 1|415 3|463 5|509 1|556 4
 -7 0| 41 2| 88 5|134 1|182 3|228 0|275 2|322 5|369 0|416 3|      510 1|557 3
 -6 0| 42 2| 89 4|135 1|183 3|229 6|276 2|323 5|370 0|417 2|464 5|511 1|558 3
 -5 0| 43 2| 90 4|136 1|      230 6|277 1|      371 0|418 2|465 4|512 1|559 3
 -4 0|       91 4|137 0|184 3|231 6|278 1|324 5|372 0|419 2|466 4|513 0|560 3
 -3 6| 44 2| 92 4|138 0|185 2|232 6|279 1|325 4|373 6|420 2|467 4|514 0|
 -2 6| 45 1| 93 3|139 0|186 2|233 5|280 1|326 4|374 6|421 1|468 4|515 0|
 -1 6| 46 1| 94 3|140 0|187 2|234 5|281 0|327 4|375 6|422 1|469 3|516 0|
  0 6| 47 1| 95 3|141 6|188 2|235 5|282 0|328 4|376 6|423 1|470 3|517 6|
  1 5| 48 1| 96 3|142 6|189 1|236 5|283 0|329 3|377 5|424 1|471 3|518 6|
  2 5| 49 0| 97 2|143 6|190 1|237 4|284 0|330 3|378 5|425 0|472 3|519 6|
  3 5| 50 0| 98 2|144 6|191 1|238 4|285 6|331 3|379 5|426 0|473 2|
  4 5| 51 0| 99 2|145 5|192 1|239 4|286 6|332 3|      427 0|474 2|520 6|
  5 4| 52 0|      146 5|193 0|      287 6|333 2|380 5|428 0|475 2|521 5|
  6 4| 53 6|100 2|147 5|194 0|240 4|288 6|334 2|381 4|429 6|476 2|522 5|

A blank entry is shown before each year number that is divisible by 28 and hyphens after each century year that's not divisible by 400 (and therefore is a non-leap year though divisible by 4).

The first thing we see is that 1960 (shown as -40) has x-value 2, not 1, as there are 11, not 10, leap years going back from 2003 to 1960.

2300 and 2500 have x-values of 3, rather than 0; I don't know where the extra discrepancy comes from. Possibly it is from considering the x-value increasing, rather than decreasing by 1 at leap years.  (The dow goes up by 2, but the y goes up by 1, and the x-value was defined as y - dow, not dow - y).

But the above table shows 84 zero x-values.

Posted by Charlie on 2006-01-22 17:13:36