tag:blogger.com,1999:blog-4115025577315673827.post4619194570928725676..comments2019-11-21T20:33:48.870+05:30Comments on CSE Blog - quant, math, computer science puzzles: "Flawless Harmony" PuzzleUnknownnoreply@blogger.comBlogger3125tag:blogger.com,1999:blog-4115025577315673827.post-62466006999668127822014-10-25T18:03:25.529+05:302014-10-25T18:03:25.529+05:30Your statement "only pair satisfying these eq...Your statement "only pair satisfying these equalities is given by (493827156, 493827165)" isn't that state forward! Can u pls elaborate its proof.<br />Yo!https://www.blogger.com/profile/15793407524283617413noreply@blogger.comtag:blogger.com,1999:blog-4115025577315673827.post-47204813707318921292014-10-23T16:06:33.507+05:302014-10-23T16:06:33.507+05:30Proof at https://www.austms.org.au/Publ/Gazette/20...Proof at https://www.austms.org.au/Publ/Gazette/2014/May14/Puzzle.pdf<br /><br />As an example, one harmonious pair is given by (123456789, 864197532).<br />If we switch the order of the last two digits, the new pair (123456798, 864197523)<br />also happens to harmonious. <br /><br />This is not a coincidence. In general, suppose we have a pair (m, n) with m = 100X + 10a + b, n = 100Y + 10c + d, where 2 ≤ a, b, c, d ≤ 9. In order for (m, n) to be harmonious, or m+n = 987654321, we must have a + c = b + d = 11. Then it is clear that the pair (m' , n') given by m' = 100X + 10b + a, n' = 100Y + 10d + c<br />also satisﬁes m' + n' = 987654321. Thus (m', n') is harmonious as well.<br /><br />In most cases, the pairs (m, n) and (m', n') are diﬀerent. The notable exception is when we have m = n' and n = m'. This can only happen if A = C, a = d and b = c. It is easy to check (by working out the digits of A = C backwards) that the only pair satisfying these equalities is given by (493827156, 493827165).<br /><br />Therefore the number of harmonious pairs must be odd.<br />Pratik Poddarhttps://www.blogger.com/profile/11577606981573330954noreply@blogger.comtag:blogger.com,1999:blog-4115025577315673827.post-72941967197416266762014-08-27T04:08:37.009+05:302014-08-27T04:08:37.009+05:30Awesome puzzle!Awesome puzzle!Alexhttps://www.blogger.com/profile/11022494336441129718noreply@blogger.com