1089.

Dominion, Volume 4, Issue 1037, 28 January 1911, Page 15

1089.

THE MYSTERIOUS NUMBERS. ' Ono more mathematical problcrh has just been solved by meii \yho.like to solve or pdricter over puKzles in figures (says the ''Wcok!y ( Scotsman"). This ono turns,upon .the, magjo,figure 10S9, vvhich.scems to possess,properties almost weird.'

Tako auy number of three, figures, ! with the only proviso that tho first and ;tho third figures be not tho same. He-. iverso thoJigurcs and snhstract the lesser 'from ;tho'''.'gi'e'!itor number. Now tako this result, reverse it and add, and the result.will- always bo 1089, no matter what number you started.' \ For-example:—Supposo you choose the number 128. . . Reverse it; you get S2l. ..:Tho latter bein" the larger, aubstract;;l2B from it. i'ou get, by substracting ;— 821 . ■ . 128 : '693 : - Now, reversing this result, wo get 390. Now add theso two 'numbers;— ', . 693 :■','■■-•'' 396 . ' ~ The answer is,lOS9 -~,... , Supposo 'we take a.'mimber ending" in Eor'o, say 990. The reverse of this is 099. So we get :- 990 '■'. ; -- ; -, 099 . Leaving .891'' This reversed is 198. Adding 891 and 198 wo again get 10S9. Supposo we try a number with a zero in the middle,. say, 102. The reverse is 201.' Subtracting, the former from tho latter we get 099. Reversing, we got, 990, which.added to 099 brings us to the mysterious 1089 .again! : 'It doesn't'matter what number wo choose at the outset. Tako 001. Reverse- it, and wo get 100. Taking 001 from lUO' wb "get 099. Reversing it comes out just the samo as the 102 example. ;

Kut why? That is a real mathematical nut to. crack. While tho solution, or rather explanation, may be made in several- ways, there are not an infinite number of them; though no two -mathe--maticiaus who havo tacked tho 1039 puzzle-have explained it in exactly tho same way. The-mystic 1089 ■is ■ really 9xll*ll. And when any number of three digits is reversed and subtracted, the lesser from the greater, tho result is invariably a multiple of 99. As in the examples shown above, 693 is 7x99; 891 is 8x99; in tho third ease tho subtraction results in.99, which.is 99x1. And'the' sequence- of these multiples of -99—that is, 99, 198, 297, 396, 495, 594, 693) 792/ 891, and-990, each added to its inverse always gives 1089. Some'havo. carried; tho problem farther, until, another mysterious number is always reached, no matter witli what you started, and this number is 79,497! Hero ends the problem, because 79,497 canuot be reversed. It makes tho samo number all over again I Hero is tho process:— 1,089 + 9,801 ' •= 10,890 ;+ 09.801 = 20,691 '+ 19,602 = 40,293 + 39,204 = 79,497 Even then the. mystery of the digits docs not ccaso if wo keep right on adding, for wo get :— 79,497 ■+ 79,497 = 158,994 ;+ 499,851 = 658,845 ;+ 548,856 = 1,207,701 + 1,077,021" '■ ■ - =2,277,722 • And when 2,277,722 is reversed we get 2,277,722!