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.