Answer :
The answer to this
quite easy puzzle may, of
course, be readily obtained by trial, deducting the largest power of 7
that is contained in one million dollars, then the next largest power
from the remainder, and so on.
But the little problem is intended to illustrate a simple direct
method.
The answer is given at once by converting 1,000,000 to the septenary
scale, and it is on this subject of scales of notation that I propose
to write a few words for the benefit of those who have never
sufficiently considered the matter.
Our manner of
figuring is a sort of perfected
arithmetical shorthand, a system devised to enable us to manipulate
numbers as rapidly and correctly as possible by means of
symbols.
If we write the number 2,341 to represent two thousand three hundred
and fortyone dollars, we wish to imply 1 dollar, added to four times
10 dollars, added to three times 100 dollars, added to two times 1,000
dollars.
From the number in the units place on the right, every figure to the
left is understood to represent a multiple of the particular power of
10 that its position indicates, while a cipher (0) must be inserted
where necessary in order to prevent confusion, for if instead of 207 we
wrote 27 it would be obviously misleading.
We thus only require ten figures, because directly a number exceeds 9
we put a second figure to the left, directly it exceeds 99 we put a
third figure to the left, and so on.
It will be seen that this is a purely arbitrary method.
It is working
in the denary (or ten) scale of notation, a system undoubtedly derived
from the fact that our forefathers who devised it had ten fingers upon
which they were accustomed to count, like our children of
today.
It is unnecessary for us ordinarily to state that we are using the
denary scale, because this is always understood in the common affairs
of life.
But if a man said
that he had 6,553 dollars in the
septenary (or seven) scale of notation, you will find that this is
precisely the same amount as 2,341 in our ordinary denary scale.
Instead of using powers of ten, he uses powers of 7, so that he never
needs any figure higher than 6, and 6,553 really stands for 3, added to
five times 7, added to five times 49, added to six times 343 (in the
ordinary notation), or 2,341.
To reverse the
operation, and convert
2,341 from the denary to the septenary scale, we divide it by 7, and
get 334 and remainder 3; divide 334 by 7, and get 47 and remainder 5;
and so keep on dividing by 7 as long as there is anything to
divide.
The remainders, read backwards, 6, 5, 5, 3, give us the answer, 6,553.
Now, as I have
said, our puzzle may be solved at
once by merely converting 1,000,000 dollars to the septenary
scale.
Keep on dividing this number by 7 until there is nothing more left to
divide, and the remainders will be found to be 11333311 which is
1,000,000 expressed in the septenary scale.
Therefore, 1 gift of 1
dollar, 1 gift of 7 dollars, 3 gifts of 49 dollars, 3 gifts of 343
dollars, 3 gifts of 2,401 dollars, 3 gifts of 16,807 dollars, 1 gift of
117,649 dollars, and one substantial gift of 823,543 dollars,
satisfactorily solves our problem.
And it is the only possible
solution.
It is thus seen that no "trials" are necessary; by converting
to the septenary scale of notation we go direct to the answer.
