But when all those combinations have been struck out that are known to be impossible, it does not follow that all the remaining “possible forms” will actually work.  The elemental form may be right enough, but there are other and deeper considerations that creep in to defeat our attempts.  For example, 98 + 2 is an impossible combination, because we are able to say at once that there is no possible form for the digital roots of the fraction equal to 2.  But in the case of 97 + 3 there is a possible form for the digital roots of the fraction, namely, 6—­5, and it is only on further investigation that we are able to determine that this form cannot in practice be obtained, owing to curious considerations.  The working is greatly simplified by a process of elimination, based on such considerations as that certain multiplications produce a repetition of figures, and that the whole number cannot be from 12 to 23 inclusive, since in every such case sufficiently small denominators are not available for forming the fractional part.

91.—­MORE MIXED FRACTIONS.

The point of the present puzzle lies in the fact that the numbers 15 and 18 are not capable of solution.  There is no way of determining this without trial.  Here are answers for the ten possible numbers:—­

9+5472/1368 = 13; 9+6435/1287 = 14; 12+3576/894 = 16; 6+13258/947 = 20; 15+9432/786 = 27; 24+9756/813 = 36; 27+5148/396 = 40; 65+1892/473 = 69; 59+3614/278 = 72; 75+3648/192 = 94.

I have only found the one arrangement for each of the numbers 16, 20, and 27; but the other numbers are all capable of being solved in more than one way.  As for 15 and 18, though these may be easily solved as a simple fraction, yet a “mixed fraction” assumes the presence of a whole number; and though my own idea for dodging the conditions is the following, where the fraction is both complex and mixed, it will be fairer to keep exactly to the form indicated:—­

      3952
      ——­
       746 = 15;
    3 ——­
        1

      5742
      ——­
       638 = 18.
    9 ——­
        1

I have proved the possibility of solution for all numbers up to 100, except 1, 2, 3, 4, 15, and 18.  The first three are easily shown to be impossible.  I have also noticed that numbers whose digital root is 8—­such as 26, 35, 44, 53, etc.—­seem to lend themselves to the greatest number of answers.  For the number 26 alone I have recorded no fewer than twenty-five different arrangements, and I have no doubt that there are many more.

92.—­DIGITAL SQUARE NUMBERS.

So far as I know, there are no published tables of square numbers that go sufficiently high to be available for the purposes of this puzzle.  The lowest square number containing all the nine digits once, and once only, is 139,854,276, the square of 11,826.  The highest square number under the same conditions is, 923,187,456, the square of 30,384.