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Full tables: "mod function" and "bitwise OR" (in the 8x8's) http://www.calcudoku.org/forum/viewtopic.php?f=3&t=84 
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Author:  jomapil [ Sat Oct 15, 2011 7:33 am ] 
Post subject:  Re: Full tables: "mod function" and "bitwise OR" (in the 8x8 
Interesting thread! And useful to those who don't know to do the operations and don't have yet the tables. And some particulars I hadn't notice. Thanks, clm. 
Author:  clm [ Mon Oct 17, 2011 4:02 pm ] 
Post subject:  Re: Full tables: "mod function" and "bitwise OR" (in the 8x8 
jomapil wrote: ... And useful to those who don't know to do the operations and don't have yet the tables...Thanks, clm. Welcome. The actual four special 8x8's puzzles ("mod function", "bitwise OR", "exponentiation" and "from 0") can be of different levels of difficulty, it is difficult to affirm that one is more difficult than the other (I am sure the difficulty can be adjusted by the program). Probably the "mod function" and the "bitwise OR" represent a bigger problem for most people because those are not very usual operations. When I joined the page, at the beginning, it was very useful for me to have a handy list of the combinations, so I prepared one. The "from 0" looks initially easier though the operations with the 0 are not "natural" and that's why some discussion took place in the past, in the Forum, with respect to multiplications and divisions (obviously, two or more 0's could be inside a cage "0x" but never two 0's can go inside a cage "0:" due to the well known "uncertainty"). I hope, then, that the tables may be useful. 
Author:  veryevilking [ Fri Apr 13, 2012 2:14 am ] 
Post subject:  Re: Full tables: "mod function" and "bitwise OR" (in the 8x8 
what about 10x10 with mods? what are the possibilities? i have the possibilities from 0 till 8 but 9 and 10 aren't included 
Author:  honkhonk [ Fri Apr 13, 2012 10:10 am ] 
Post subject:  Re: Full tables: "mod function" and "bitwise OR" (in the 8x8 
Author:  clm [ Fri Apr 13, 2012 2:25 pm ] 
Post subject:  Re: Full tables: "mod function" and "bitwise OR" (in the 8x8 
veryevilking wrote: what about 10x10 with mods? what are the possibilities? i have the possibilities from 0 till 8 but 9 and 10 aren't included When the tables were made (six months ago and in fact until this moment) the main page was only showing the mod function in puzzles up to 8x8 (actually in an 8x8 and a 7x7 "manyop" every week). I understand that, for intance, in the books, they appear larger puzzles with the mod function, even in the 12x12 format. Apart of the recent very handy table just provided by honkhonk, I'm sending in a separate post an Appendix to the original tables going now to 12x12 (the philosophy is the same). 
Author:  clm [ Fri Apr 13, 2012 2:33 pm ] 
Post subject:  Appendix 1: “mod function” full tables (up to 12x12’s). 
Appendix 1: “mod function” full tables (up to 12x12’s). Here is the full table for the “mod function” (the number of combinations for each case is shown in brackets). The “mod” function, xmod, works dividing the two numbers inside the cage, considering any of them as the dividend and the other as the divisor, thus obtaining a quotient and a remainder x, this remainder is considered the result of the operation, for instance, in 4 / 11, the quotient is 0, the remainder is 4, so the pair 411 is a valid solution for a 4mod (among others); but considering 11 / 4 (the “natural” division), the quotient is 2, and the remainder is 3, so the pair 411 is a valid solution for 3mod (among others). Mod: Here is the full table (mistakes reported welcome): 11mod: 1112 (1 combination) 10mod: 1011 1012 (2) 9mod: 910 911 912 (3) 8mod: 89 810 811 812 (4) 7mod: 78 79 710 711 712 (5) 6mod: 67 68 69 610 611 612 (6) 5mod: 56 57 58 59 510 511 512 and additionally 127 116 (9) 4mod: 45 46 47 48 49 410 411 412 and 128 117 106 95 (12) 3mod: 34 35 36 37 38 39 310 311 312 and 129 118 114 107 96 85 74 (16) 2mod: 23 24 25 26 27 28 29 210 211 212 and 1210 125 119 113 108 104 97 86 83 75 64 53 (22) 1mod: 12 13 14 15 16 17 18 19 110 111 112 and 1211 1110 115 109 103 98 94 92 87 76 73 72 65 54 52 43 32 (28) 0mod: 12 13 14 15 16 17 18 19 110 111 112 and 126 124 123 122 105 102 93 84 82 63 62 42 (23). Total of possible combinations: 131. The main tips are: 1) A number x can never be inside a cage ymod if x < y, for instance, 2 can never go inside a cage 3mod, 4mod, 5mod, 6mod, … . 2) In a cage 0mod or 1mod "any" number (from 1 to 12) can initially be present. 3) In cages 6mod, 7mod, 8mod, 9mod, 10mod and 11mod, the 6, 7, 8, 9, 10 and 11, respectively, must be present; the cages 1mod, 2mod, 3mod, 4mod and 5mod are possible without the respective 1, 2, 3, 4 or 5. The cage 0mod has pairs that are multiples or a 1 is inside, what in fact is a particular case of the multiples. 
Author:  marblevolcano [ Thu Sep 15, 2016 1:44 am ] 
Post subject:  Re: Full tables: "mod function" and "bitwise OR" (in the 8x8 
I'm just going to comment on this to keep it at the top of the "Solving Strategies and Tips" section for reference. 
Author:  marblevolcano [ Thu Sep 15, 2016 1:46 am ] 
Post subject:  Re: Full tables: "mod function" and "bitwise OR" (in the 8x8 
Also, could you have a mod cage with 3 or more cells? For example, 7mod4=3mod2=1 (This could be entirely wrong, but oh well ) which would be 1mod and contain 7, 4, and 2? 
Author:  clm [ Thu Sep 15, 2016 1:21 pm ] 
Post subject:  Re: Full tables: "mod function" and "bitwise OR" (in the 8x8 
marblevolcano wrote: Also, could you have a mod cage with 3 or more cells? For example, 7mod4=3mod2=1 (This could be entirely wrong, but oh well ) which would be 1mod and contain 7, 4, and 2? I agree, also in my opinion there is nothing wrong with iterating the mod function (and in fact it would be an interesting variation). The reason why I havenn't calculate the combinations for 3cell (or more) cages is because since the beginning of the site, in 2009, Patrick has never proposed a puzzle where the mod function appeared in a cage bigger than 2 cells (including books, though the original intention of this post was to help in the site itself and not to cover all books later published, in fact, in the books, they have appeared wider than 8x8 puzzles with those functions). As soon as the mod function is extended I would try to extend the tables for those situations. Similarly I havenn't seen any cage bigger than 3cell with the Bitwise OR function. (The exponentiation or subtraction cages require to "start" in a particular operand to obtain the final result, that is, some "sequence" is required in the operands while with the mod or Bitwise OR cages this is not necessary.) 
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