MORE MIXED FRACTIONS.
(
Money Puzzles)
When I first published my solution to the last puzzle, I was led to
attempt the expression of all numbers in turn up to 100 by a mixed
fraction containing all the nine digits. Here are twelve numbers for the
reader to try his hand at: 13, 14, 15, 16, 18, 20, 27, 36, 40, 69, 72,
94. Use every one of the nine digits once, and only once, in every case.
Answer:
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.
