The problem of expressing the number 100 as a mixed number or fraction,
using all the nine digits once, and once only, has, like all these
digital puzzles, a fascinating side to it. The merest tyro can by
patient trial obtain correct results, and there is a singular pleasure
in discovering and recording each new arrangement akin to the delight of
the botanist in finding some long-sought plant. It is simply a matter of
arranging those nine figures correctly, and yet with the thousands of
possible combinations that confront us the task is not so easy as might
at first appear, if we are to get a considerable number of results. Here
are eleven answers, including the one I gave as a specimen:--
2148 1752 1428 1578
96 ----, 96 ----, 96 ----, 94 ----,
537 438 357 263
7524 5823 5742 3546
91 ----, 91 ----, 91 ----, 82 ----,
836 647 638 197
7524 5643 69258
81 ----, 81 ----, 3 -----.
396 297 714
Now, as all the fractions necessarily represent whole numbers, it will
be convenient to deal with them in the following form: 96 + 4, 94 + 6,
91 + 9, 82 + 18, 81 + 19, and 3 + 97.
With any whole number the digital roots of the fraction that brings it
up to 100 will always be of one particular form. Thus, in the case of 96
+ 4, one can say at once that if any answers are obtainable, then the
roots of both the numerator and the denominator of the fraction will be
6. Examine the first three arrangements given above, and you will find
that this is so. In the case of 94 + 6 the roots of the numerator and
denominator will be respectively 3--2, in the case of 91 + 9 and of 82 +
18 they will be 9--8, in the case of 81 + 19 they will be 9--9, and in
the case of 3 + 97 they will be 3--3. Every fraction that can be
employed has, therefore, its particular digital root form, and you are
only wasting your time in unconsciously attempting to break through this
law.
Every reader will have perceived that certain whole numbers are
evidently impossible. Thus, if there is a 5 in the whole number, there
will also be a nought or a second 5 in the fraction, which are barred by
the conditions. Then multiples of 10, such as 90 and 80, cannot of
course occur, nor can the whole number conclude with a 9, like 89 and
79, because the fraction, equal to 11 or 21, will have 1 in the last
place, and will therefore repeat a figure. Whole numbers that repeat a
figure, such as 88 and 77, are also clearly useless. These cases, as I
have said, are all obvious to every reader. But when I declare that such
combinations as 98 + 2, 92 + 8, 86 + 14, 83 + 17, 74 + 26, etc., etc.,
are to be at once dismissed as impossible, the reason is not so evident,
and I unfortunately cannot spare space to explain it.
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.