THE MILLIONAIRE'S PERPLEXITY.
(
Money Puzzles)
Mr. Morgan G. Bloomgarten, the millionaire, known in the States as the
Clam King, had, for his sins, more money than he knew what to do with.
It bored him. So he determined to persecute some of his poor but happy
friends with it. They had never done him any harm, but he resolved to
inoculate them with the "source of all evil." He therefore proposed to
distribute a million dollars among them and watch them go rapidly to the
bad. But he was a man of strange fancies and superstitions, and it was
an inviolable rule with him never to make a gift that was not either one
dollar or some power of seven--such as 7, 49, 343, 2,401, which numbers
of dollars are produced by simply multiplying sevens together. Another
rule of his was that he would never give more than six persons exactly
the same sum. Now, how was he to distribute the 1,000,000 dollars? You
may distribute the money among as many people as you like, under the
conditions given.
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 forty-one 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 to-day. 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.
