jasen 4 Posted February 4, 2016 Report Share Posted February 4, 2016 (edited) Find the maximum number of distinct ways you can create combination of six digit number (from 0 through 9). with following rules : 1. Each number can be used more than one. 2. Any two combinations can have at most 1 digit in same place. example : a. 075922 and 166433 is accepted. b. 075922 and 166432 is accepted, both put 2 in 6th place). c. 075922 and 066432 cannot both be used, because both put 0 in first place, and 2 in 6th place). Edited February 4, 2016 by jasen Quote Link to post Share on other sites

0 Solution DejMar 9 Posted February 4, 2016 Solution Report Share Posted February 4, 2016 Spoiler As each 6-decimal-digit string can have at most one digit shared positionally between any two strings of digits, there is at most ten 6-decimal-digit strings that can occur for any given position. The answer to the problem is then the same for any decimal-digit string of length n, such that n > 1, which is 10^{2} = 100 strings of n-decimal-digits. The number of different sets of one hundred 6-decimal-digit strings that can be formed is much greater finite number, but does not change the answer to the total number of different ways one can create a combination of six digit numbers, using the decimal digits within the given constraints. Quote Link to post Share on other sites

1 Logophobic 9 Posted February 5, 2016 Report Share Posted February 5, 2016 I think DejMar was right, though I have not yet proven it. Here is my solution for N=3 (100): Spoiler 000 101 202 303 404 505 606 707 808 909 011 112 213 314 415 516 617 718 819 910 022 123 224 325 426 527 628 729 820 921 033 134 235 336 437 538 639 730 831 932 044 145 246 347 448 549 640 741 842 943 055 156 257 358 459 550 651 752 853 954 066 167 268 369 460 561 662 763 864 965 077 178 279 370 471 572 673 774 875 976 088 189 280 381 482 583 684 785 886 987 099 190 291 392 493 594 695 796 897 998 Quote Link to post Share on other sites

0 bonanova 84 Posted February 4, 2016 Report Share Posted February 4, 2016 Unless I'm counting wrong, it's Spoiler digits accepted permutations 2 100 3 76 4 70 5 46 6 62 Quote Link to post Share on other sites

0 jasen 4 Posted February 4, 2016 Author Report Share Posted February 4, 2016 (edited) @bonanova. How do you get the numbers? My answer is very close to your answer for 6 digits numbers. If I search the numbers by computer start from 000000 to 999999 the result is exactly same with your calculation. (although maybe it is not the best answer) ["000000","011111","022222","033333","044444","055555","066666","077777","088888","099999","101234","110325","123016", "132107","145670","154761","167452","176543","202318","213209","220153","231042","246735","257624","264580","275491","303145","312054", "321360","330271","347506","356417","365723","374632","404826","415937","428405","439514","480649","491758","506951","517840","529687","538796", "581473","590562","608579","619468","626894","637985","684257","695346","724978","735869","748021","759130","786302","797213","842963", "853872","962831","973920"] Edited February 4, 2016 by jasen Quote Link to post Share on other sites

0 bonanova 84 Posted February 4, 2016 Report Share Posted February 4, 2016 Same approach. May not be sophisticated, likely not optimally efficient, but took only about 15 minutes. We're both "very close" and "exactly the same"? I guess we haven't proved the answers are unique. Starting with a different seed might yield different results.Anyway, here are my numbers: Spoiler N=2 (all 100, obviously) 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 N=3 (76)000 011 022 033 044 055 066 077 088 099 101 110 123 132 145 154 167 176 189 198 202 213 220 231 246 257 264 275 303 312 321 330 347 356 365 374 404 415 426 437 440 451 462 473 505 514 527 536 541 550 563 572 606 617 624 635 642 653 660 671 707 716 725 734 743 752 761 770 808 819 880 891 909 918 981 990 N=4 (70)0000 0111 0222 0333 0444 0555 0666 0777 0888 0999 1012 1103 1230 1321 1456 1547 1674 1765 2023 2132 2201 2310 2467 2576 2645 2754 3031 3120 3213 3302 3475 3564 3657 3746 4048 4159 4284 4395 4806 4917 5069 5178 5296 5387 5814 5905 6085 6194 6249 6358 6827 6936 7097 7186 7268 7379 7835 7924 8408 8519 8680 8791 8842 8953 9429 9538 9692 9783 9850 9941 N=5 (46)00000 01111 02222 03333 04444 05555 06666 07777 08888 09999 10123 11032 12301 13210 14567 15476 16745 17654 20231 21320 22013 23102 24675 25764 26457 27546 30312 31203 32130 33021 34756 35647 36574 37465 40489 41598 48904 49815 50894 51985 58049 59158 60948 61859 68195 69084 N=6 (62)000000 011111 022222 033333 044444 055555 066666 077777 088888 099999 101234 110325 123016 132107 145670 154761 167452 176543 202318 213209 220153 231042 246735 257624 264580 275491 303145 312054 321360 330271 347506 356417 365723 374632 404826 415937 428405 439514 480649 491758 506951 517840 529687 538796 581473 590562 608579 619468 626894 637985 684257 695346 724978 735869 748021 759130 786302 797213 842963 853872 962831 973920 Quote Link to post Share on other sites

## Question

## jasen 4

Find the maximum number of distinct ways you can create combination of six digit number

(from 0 through 9). with following rules :

1. Each number can be used more than one.

2. Any two combinations can have at most 1 digit in same place.

example :

a. 075922 and 166433 is accepted.

b. 075922 and 166432 is accepted, both put 2 in 6th place).

c. 075922 and 066432 cannot both be used,

because both put 0 in first place, and 2 in 6th place).

Edited by jasen## Link to post

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