Answer :
The sum of all the
numbers that can be formed with
any given set of four different figures is always 6,666 multiplied by
the sum of the four figures. Thus, 1, 2, 3, 4 add up 10, and ten times
6,666 is 66,660. Now, there are thirty-five different ways of selecting
four figures from the seven on the dice—remembering the 6 and
9 trick. The figures of all these thirty-five groups add up to 600.
Therefore 6,666 multiplied by 600 gives us 3,999,600 as the correct
answer.
Let us discard the
dice and deal with the problem
generally, using the nine digits, but excluding nought. Now, if you
were given simply the sum of the digits—that is, if the
condition were that you could use any four figures so long as they
summed to a given amount—then we have to remember that
several combinations of four digits will, in many cases, make the same
sum.
|
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
|
1 |
1 |
2 |
3 |
5 |
6 |
8 |
9 |
11 |
11 |
12 |
| |
|
21 |
22 |
23 |
24 |
25 |
26 |
27 |
28 |
29 |
30 |
|
11 |
11 |
9 |
8 |
6 |
5 |
3 |
2 |
1 |
1 |
Here the top row of
numbers gives all the possible
sums of four different figures, and the bottom row the number of
different ways in which each sum may be made. For example 13 may be
made in three ways: 1237, 1246, and 1345. It will be found that the
numbers in the bottom row add up to 126, which is the number of
combinations of nine figures taken four at a time. From this table we
may at once calculate the answer to such a question as this: What is
the sum of all the numbers composed of our different digits (nought
excluded) that add up to 14? Multiply 14 by the number beneath t in the
table, 5, and multiply the result by 6,666, and you will have the
answer. It follows that, to know the sum of all the numbers composed of
four different digits, if you multiply all the pairs in the two rows
and then add the results together, you will get 2,520, which,
multiplied by 6,666, gives the answer 16,798,320.
The following
general solution for any number of
digits will doubtless interest readers. Let n represent number of
digits, then 5 (10n - 1)
) 8!
divided by (9 - n)! equals the required sum. Note
that 0! equals 1. This may be reduced to the following practical rule:
Multiply together
4 × 7 × 6 × 5
... to (n - 1) factors; now add
(n + 1) ciphers to the right, and from this result
subtract the same set of figures with a single cipher to the right.
Thus for n = 4 (as in the case last mentioned),
4 × 7 × 6 = 168.
Therefore 16,800,000 less 1,680 gives us 16,798,320 in another way.
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