6729/13458 = 1/2

    5823/17469 = 1/3

    3942/15768 = 1/4

    2697/13485 = 1/5

    2943/17658 = 1/6

    2394/16758 = 1/7

    3187/25496 = 1/8

    6381/57429 = 1/9

The sum of the numerator digits and the denominator digits will, of course, always be 45, and the “digital root” is 9.  Now, if we separate the nine digits into any two groups, the sum of the two digital roots will always be 9.  In fact, the two digital roots must be either 9—­9, 8—­1, 7—­2, 6—­3, or 5—­4.  In the first case the actual sum is 18, but then the digital root of this number is itself 9.  The solutions in the cases of one-third, one-fourth, one-sixth, one-seventh, and one-ninth must be of the form 9—­9; that is to say, the digital roots of both numerator and denominator will be 9.  In the cases of one-half and one-fifth, however, the digital roots are 6—­3, but of course the higher root may occur either in the numerator or in the denominator; thus 2697/13485, 2769/13845, 2973/14865, 3729/18645, where, in the first two arrangements, the roots of the numerator and denominator are respectively 6—­3, and in the last two 3—­6.  The most curious case of all is, perhaps, one-eighth, for here the digital roots may be of any one of the five forms given above.

The denominators of the fractions being regarded as the numerators multiplied by 2, 3, 4, 5, 6, 7, 8, and 9 respectively, we must pay attention to the “carryings over.”  In order to get five figures in the product there will, of course, always be a carry-over after multiplying the last figure to the left, and in every case higher than 4 we must carry over at least three times.  Consequently in cases from one-fifth to one-ninth we cannot produce different solutions by a mere change of position of pairs of figures, as, for example, we may with 5832/17496 and 5823/17469, where the 2/6 and 3/9 change places.  It is true that the same figures may often be differently arranged, as shown in the two pairs of values for one-fifth that I have given in the last paragraph, but here it will be found there is a general readjustment of figures and not a simple changing of the positions of pairs.  There are other little points that would occur to every solver—­such as that the figure 5 cannot ever appear to the extreme right of the numerator, as this would result in our getting either a nought or a second 5 in the denominator.  Similarly 1 cannot ever appear in the same position, nor 6 in the fraction one-sixth, nor an even figure in the fraction one-fifth, and so on.  The preliminary consideration of such points as I have touched upon will not only prevent our wasting a lot of time in trying to produce impossible forms, but will lead us more or less directly to the desired solutions.

89.—­ADDING THE DIGITS.

The smallest possible sum of money is L1, 8s. 93/4d., the digits of which add to 25.

90.—­THE CENTURY PUZZLE.