tag:blogger.com,1999:blog-5730391.post257551896177886905..comments2019-08-19T13:32:55.741-07:00Comments on Blaine's Puzzle Blog: How about a math puzzle?Blainehttp://www.blogger.com/profile/06379274325110866036noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-5730391.post-23457452558199316582008-02-03T00:21:00.000-08:002008-02-03T00:21:00.000-08:00Indeed, your answer solves the puzzle:2 x 19 = 383...Indeed, your answer solves the puzzle:<BR/><BR/>2 x 19 = 38<BR/>3 x 127 = 381<BR/>4 x 954 = 3816<BR/>5 x 7633 = 38165<BR/>6 x 63609 = 381654<BR/>7 x 545221 = 3816547<BR/>8 x 4770684 = 38165472<BR/>9 x 42406081 = 381654729<BR/><BR/>However, you can get there with logic rather than a brute force program:<BR/><BR/>We know that ABCDE must be a multiple of 5, so E must be 5.<BR/><BR/>AB, ABCD, ABCDEF, ABCDEFGH are multiples of 2, 4, 6 and 8 respectively. Thus their last digits (B, D, F and H) must be even. That leaves 1, 3, 7 and 9 for A, C, G and I.<BR/><BR/>ABCD is a multiple of 4, so CD must be a multiple of 4. Given that C is odd, the possible values are {12, 16, 32, 36, 72, 76, 92 and 96}. That means D must be 2 or 6.<BR/><BR/>Similarly ABCDEFGH is a multiple of 8 (and therefore 4). Using similar logic, GH will be in {12, 16, 32, 36, 72, 76, 92 and 96}. Therefore H must be 2 or 6.<BR/><BR/>Because 2 and 6 are accounted for, B and F must be 4 and 8. That means FGH will be {412, 416, 432, 436, 472, 476, 492, 496, 812, 816, 832, 836, 872, 876, 892 or 896}. <BR/><BR/>But FGH needs to be a multiple of 8 as well as a multiple of 4. The only multiples of 8 are {416, 472, 496, 816, 872 or 896}<BR/><BR/>ABC is a multiple of 3 so its digits must add to be a multiple of 3. Possible choices are {147, 183, 189, 381, 387, 741, 783, 789, 981, 987}<BR/><BR/>ABCDEF is a multiple of 6 and therefore also a multiple of 3. Given that ABC already adds to a multiple of 3, we only need to check DEF. E is 5. Possible choices for DEF are {258, 654}<BR/><BR/>Putting things together we have a couple possible patterns:<BR/>A-4-C-2-5-8-G-6-I<BR/>or<BR/>A-8-C-6-5-4-G-2-I<BR/><BR/>That reduces the set of choices for FGH to {472, 816, 896}<BR/><BR/>So the 3 possible patterns are now:<BR/>A-4-C-2-5-8-1-6-I<BR/>A-4-C-2-5-8-9-6-I<BR/>A-8-C-6-5-4-7-2-I<BR/><BR/>Looking at our set of possible values for ABC we have {147, 183, 189, 381, 387, 741, 783, 789, 981, 987}. If B = 4, then ABC is either {147 or 741}. That eliminates the first string. If B = 8, all the values using 7 are eliminated.<BR/><BR/>So now we have:<BR/><BR/>{147 or 741}<BR/>1-4-7-2-5-8-9-6-I<BR/>7-4-1-2-5-8-9-6-I<BR/><BR/>{183, 189, 381, 981}<BR/>1-8-3-6-5-4-7-2-I<BR/>1-8-9-6-5-4-7-2-I<BR/>3-8-1-6-5-4-7-2-I<BR/>9-8-1-6-5-4-7-2-I<BR/><BR/>Okay there are 6 sequences to check to see if ABCDEFG is a multiple of 7.<BR/>1472589 / 7 = 210369.857142...<BR/>7412589 / 7 = 1058941.285714...<BR/>1836547 / 7 = 262363.857142...<BR/>1896547 / 7 = 270935.285714...<BR/>3816547 / 7 = 545221<BR/>9816547 / 7 = 1402363.857142...<BR/><BR/>As you can see only one is evenly divisable by 7. <BR/>3816547<BR/><BR/>That's part of this pattern:<BR/>3-8-1-6-5-4-7-2-I<BR/><BR/>The final digit is 9. Obviously the nine digits add up to 45, so the full number will be a multiple of nine regardless.<BR/><BR/>Answer:<BR/>381654729Blainehttps://www.blogger.com/profile/06379274325110866036noreply@blogger.comtag:blogger.com,1999:blog-5730391.post-18346257659984782392008-02-01T15:42:00.000-08:002008-02-01T15:42:00.000-08:00I couldn't figure out a shortcut to determining th...I couldn't figure out a shortcut to determining the number, so I ended up making a computer program to do the work for me.<BR/><BR/>Though crudely written, it did find the answer for me.<BR/><BR/>The number is 381654729Ralphhttps://www.blogger.com/profile/16194974915716525500noreply@blogger.com