Could They Be The Same Day Of The Week?

Question

Challenge

Given a non-negative integer, output whether it is possible for two dates (of the Gregorian calendar) differing by exactly that many years to share a day of the week. A year is assumed to be a leap year either if it is divisible by 4 but not by 100, or if it is divisible by 400.

Output may be:

• falsey/truthy (in either orientation)
• any two distinct values
• one distinct value and one being anything else
• by program return code
• by success/error
• by any other reasonable means - just ask if you suspect it may be controversial

But not by two non-distinct sets of values, except for falsey/truthy (as this would allow a no-op!)

Detail

This is whether the input is a member of OEIS sequence A230995.

Members:

0, 5, 6, 7, 11, 12, 17, 18, 22, 23, 28, 29, 33, 34, 35, 39, 40, 45, 46, 50, 51, 56, 57, 61, 62, 63, 67, 68, 73, 74, 78, 79, 84, 85, 89, 90, 91, 95, 96, 101, 102, 106, 107, 108, 112, 113, 114, 117, 118, 119, 123, 124, 125, 129, 130, 131, 134, 135, 136, 140, 141, 142, 145, 146, 147, 151, 152, 153, 157, 158, 159, 162, 163, 164, 168, 169, 170, 173, 174, 175, 179, 180, 181, 185, 186, 187, 190, 191, 192, 196, 197, 198, 202, 203, 204, 208, 209, 210, 213, 214, 215, 219, 220, 221, 225, 226, 227, 230, 231, 232, 236, 237, 238, 241, 242, 243, 247, 248, 249, 253, 254, 255, 258, 259, 260, 264, 265, 266, 269, 270, 271, 275, 276, 277, 281, 282, 283, 286, 287, 288, 292, 293, 294, 298, 299, 304, 305, 309, 310, 311, 315, 316, 321, 322, 326, 327, 332, 333, 337, 338, 339, 343, 344, 349, 350, 354, 355, 360, 361, 365, 366, 367, 371, 372, 377, 378, 382, 383, 388, 389, 393, 394, 395
plus
400, 405, 406, 407, 411, ...

Non-members:

1, 2, 3, 4, 8, 9, 10, 13, 14, 15, 16, 19, 20, 21, 24, 25, 26, 27, 30, 31, 32, 36, 37, 38, 41, 42, 43, 44, 47, 48, 49, 52, 53, 54, 55, 58, 59, 60, 64, 65, 66, 69, 70, 71, 72, 75, 76, 77, 80, 81, 82, 83, 86, 87, 88, 92, 93, 94, 97, 98, 99, 100, 103, 104, 105, 109, 110, 111, 115, 116, 120, 121, 122, 126, 127, 128, 132, 133, 137, 138, 139, 143, 144, 148, 149, 150, 154, 155, 156, 160, 161, 165, 166, 167, 171, 172, 176, 177, 178, 182, 183, 184, 188, 189, 193, 194, 195, 199, 200, 201, 205, 206, 207, 211, 212, 216, 217, 218, 222, 223, 224, 228, 229, 233, 234, 235, 239, 240, 244, 245, 246, 250, 251, 252, 256, 257, 261, 262, 263, 267, 268, 272, 273, 274, 278, 279, 280, 284, 285, 289, 290, 291, 295, 296, 297, 300, 301, 302, 303, 306, 307, 308, 312, 313, 314, 317, 318, 319, 320, 323, 324, 325, 328, 329, 330, 331, 334, 335, 336, 340, 341, 342, 345, 346, 347, 348, 351, 352, 353, 356, 357, 358, 359, 362, 363, 364, 368, 369, 370, 373, 374, 375, 376, 379, 380, 381, 384, 385, 386, 387, 390, 391, 392, 396, 397, 398, 399
plus
401, 402, 403, 404, 408, ...

This is code-golf so the shortest answer in each language wins!

Answer 1

Python 2, 58 bytes

u=-abs(200-input()%400)-4
print u/100+5>(u-8)*5/4%7>u%4/-3

Try it online!

A direct formula.

Answer 2

Jelly, 20 18 bytes

99R4ḍj`‘ṡ%4ȷ$S€P7ḍ

Outputs 1 for members, 0 for non-members.

Try it online!

How it works

99R4ḍj`‘ṡ%4ȷ$S€P7ḍ  Main link. Argument: n

99                  Set the return value to 99.
  R                 Range; yield [01, .., 99].
   4ḍ               Test each element in the range for divisibility by 4.
     j`             Join the resulting array, using itself as separator.
                    The result is an array of 9801 Booleans indicating whether the
                    years they represent have leap days.
       ‘            Increment the results, yielding 1 = 365 (mod 7) for non-leap
                    years, 2 = 366 (mod 7) for leap years.
         %4ȷ$       Compute n % 4000.
        ṡ           Take all slices of length n % 4000 of the result to the left.
             S€     Take the sum of each slice.
               P    Take the product of the sums.
                7ḍ  Test for divisibility by 7.

Answer 3

MATL, 17 bytes

`0Gv@+5:YcYO8XOda

The program halts if the input belongs to the sequence, or runs indefinitely (infinite loop) otherwise.

Let n be the input. The code executes a loop that tests years 1 and 1+n; then 2 and 2+n; ... until a matching day of the week is found. If no matching exists the loop runs indefinitely.

The membership function for n is periodic with period 400. Therefore, at most 400 iterations are needed if n belongs to the sequence. This requires less than 20 seconds in Try It Online. As a proof of this upper bound, here's a modified program that limits the number of iterations to 400 (by adding @401<* at the end). Note also that this bound is loose, and a few seconds usually suffice.

Try it online!

Explanation

`           % Do...while
  0Gv       %   Push column vector [0; n], where n is the input number
  @+        %   Add k, element-wise. Gives [k; k+n]
  5:        %   Push row vector [1, 2, 3, 4, 5]
  Yc        %   Horizontal "string" concatenation: gives the 2×6 matrix
            %   [k, 1, 2, 3, 4, 5; k+n, 1, 2, 3, 4, 5]. The 6 columns
            %   represent year, month, day, hour, minute, second
  YO        %   Convert each row to serial date number. Gives a column
            %   vector of length 2
  8XO       %   Convert each date number to date string with format 8,
            %   which is weekday in three letters ('Mon', 'Tue', etc).
            %   This gives a 2×3 char matrix such as ['Wed';'Fri']
  d         %   Difference (of codepoints) along each column. Gives a
            %   row vector of length 3
  a         %   True if some element is nonzero, or false otherwise
            % End (implicit). The loop proceeds with the next iteration
            % if the top of the stack is true

---

Old version, 24 bytes

400:"0G&v@+5:YcYO8XOdavA

Output is 0 if the input belongs to the sequence, or 1 otherwise.

Try it online!

Explanation

400         % Push row vector [1, 2, ..., 400]
"           % For each k in that array
  0G&v      %   Push column vector [0; n], where n is the input number
  @+        %   Add k, element-wise. Gives [k; k+n]
  5:        %   Push row vector [1, 2, 3, 4, 5]
  Yc        %   Horizontal "string" concatenation: gives the 2×6 matrix
            %   [k, 1, 2, 3, 4, 5; k+n, 1, 2, 3, 4, 5]. The 6 columns
            %   represent year, month, day, hour, minute, second
  YO        %   Convert each row to serial date number. Gives a column
            %   vector of length 2
  8XO       %   Convert each date number to date string with format 8,
            %   which is weekday in three letters ('Mon', 'Tue', etc).
            %   This gives a 2×3 char matrix such as ['Wed';'Fri']
  d         %   Difference (of codepoints) along each column. Gives a
            %   row vector of length 3
  a         %   True if some element is nonzero, or false otherwise
  v         %   Concatenate vertically with previous results
  A         %   True if all results so far are true
            % End (implicit). Display (implicit)

Answer 4

Python 2, 83 bytes

lambda i:all(c(y)-c(y-i)for y in range(401))
c=lambda n:(5*(n/4)+n%4-n/100+n/400)%7

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Direct port of my Haskell answer.

Answer 5

Haskell, 76 bytes

-35 bytes thanks to Jonathan Allan. -2 bytes thanks to Lynn.

f i=or[c y==c$y+i|y<-[0..400]]
c n=(5*n#4+n%4-n#100+n#400)%7
(%)=mod
(#)=div

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Using the OEIS PARI program's algorithm.

Answer 6

Pyth, 32 bytes

iI7*FsM.:hMs.iKiIL4S99*98]K%Q400

Try it here! (Click "Switch to Test Suite" to verify more test cases at once)

How?

Uses a cool trick I just added to the "Tips for golfing in Pyth" thread.

iI7*FsM.:hMs.iKiIL4S99*98]K%Q400 | Full program. Reads from STDIN, outputs to STDOUT.

                   S99           | Generate the integers in 1 ... 99.
                 L               | For each integer N in that list...
               iI 4              | Check if 4 is invariant over applying GCD with N.
                                 | This is equivalent to checking if 4 | N.
              K                  | Store the result in a variable K.
            .i         *98]K     | And interleave K with the elements of K wrapped
                                 | into a list and repeated 98 times.
           s                     | Flatten.
         hM                      | Increment.
       .:                        | And generate all the substrings...
                           %Q400 | Of length  % 400.
     sM                          | Sum each.
   *F                            | And apply folded product.
iI7                              | Check if 7 is invariant when applied GCD with the
                                 | product (basically check whether 7 | product).
                                 | Implicitly output the appropriate boolean value.

Answer 7

Python 3, 110 107 bytes

from datetime import*
lambda n,d=date:0in[d(i,1,1).weekday()-d(i+n%400,1,1).weekday()for i in range(1,999)]

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-3 bytes thanks to Mr. Xcoder.