Four Digit Sum

Considering the numbers that starts with the digit $1$ ,

$1234$

$1243$

$1324$

$1342$

$1423$

$1432$

$\text{and so on...}$

We can see that $1$ digit appeared in the first column $6$ times, continuing this sequence, we can say therefore that each digit appears in each column $6$ times. So each column must add up to $6(1+2+3+4)=60$

Therefore, the total is

$60(1000)+60(100)+60(10)+60(1)=60(1000+100+10+1)=60(1111)=66660$

Fro each one of the combinations there is exactly one reversed, so $(1234 + 4321) \times 12 = 66660$ .

Edit:Look at comment for the correct solution.

Each digit can occupy each place 6 times.

Hence, our sum can be calculated as:

$6 × (1 + 2 + 3 + 4) × (1000 + 100 + 10 + 1) =$

$= 6 × 10 × 1111 = \boxed {66660}$

If we pair off each number so that each pair equals 5555, 1234+4321 gives 5555. As there are 12 pairs of these numbers that give 5555, 5555*12 = 66660

Let be (a), (a+1), (a+2) and (a+3) 4 following digits.

The number LaTex: $(a)(a+1)(a+2)(a+3)$ is equal to

LaTeX: $a10^{3}+(a+1)10^{2}+(a+2)10^{1}+(a+3)10^{0}$

The number of permutations of the 4 digits are :

LaTeX: $A^4_4 = P_4 = 4! = 4 \times 3 \times 2 \times 1 = {\color{#D61F06} 24}$

In each rank LaTeX: $10^{n}$ , we have LaTeX: $P_4 \times a$ or LaTex: ${\color{#D61F06}24 \times a }$ and LaTex: ${\color{#3D99F6}6 \times (S_4) }$

LaTeX: $S_4 = 6\displaystyle \sum_{j=0}^{j=3}j = S_3 = 6\displaystyle \sum_{j=1}^{j=3}j = 6 \times (1+2+3) = 6 \times\frac {3 \times 4}{2} = {\color{#3D99F6}36}$

We get taking into account each rank :

LaTeX: $\displaystyle \sum_{i=1}^{i=4}[10^{(i-1)}[{\color{#D61F06}24a} + \displaystyle \sum_{j=1}^{j=3}j ]]= \displaystyle \sum_{i=1}^{i=4}[10^{(i-1)}[{\color{#D61F06}24a} + {\color{#3D99F6}36} ]] = [{\color{#D61F06}24a} + {\color{#3D99F6}36} ][ \displaystyle \sum_{i=1}^{i=4}[10^{(i-1)}]$

LaTeX: $({\color{#D61F06}24a} + {\color{#3D99F6}36})(10^{0} + 10^{1} + 10^{2} + 10^{3}) = ({\color{#D61F06}24a} + {\color{#3D99F6}36})(1 + 10 + 100 + 1000)$

LaTeX: $({\color{#D61F06}24a} + {\color{#3D99F6}36})(1111) = \boxed{26664a + 39996}$

In the case we are interested in, LaTeX: $a = 1$

And at last, we get : LaTeX: $26664 + 39996 = {\color{blueviolet}\boxed{66660}}$

The Gaussian insight is to realize that each column, ones, tens, hundreds, and thousands contain the same digits which are 6 ones, 6 twos, 6 threes, and 6 fours even though the order of the digits will be different in each column.

$(6\times 1) +( 6\times 2)+ (6 \times 3)+ ( 6\times 4)= 60$ .

The total sum is $60+600+6000+60,000= 66660$ .

There are 24 combinations of using the digits $1, 2, 3, 4$ exactly once. There are $4$ possible numbers per digit with equal probability for the thousands, hundreds, tens and ones digits, therefore, there are $\frac{24}{4}$ or $6$ numbers with each number in each digit. For example, there are 6 numbers with 1 in the ones digit, 6 numbers with 4 in the tens digit etc. The sum of all $24$ numbers of each digit are $6+6(2)+6(3)+6(4) = 60.$ Doing the addition gets us $60+600+6000+60000 = \boxed{66660}$

Firstly, we all know that $4*6$ equals to 24, $3*6$ equals to 18, $2*6$ equals to 12 and $1*6$ equals to 6. Using those numbers, we add them all up so that it equals to 60. Next we take the 60 and add that to 600 then 6000 then 60000 which should get the answer 66660. The reason we added an extra 0 is because we are actually adding the tens column then the hundreds then the thousands and so on. P.S i hope this make sense! :)

Average the maximum and minimum values, then multiply by 24:

$\frac{(1234 + 4321)}{2}$ = $\frac{5555}{2}$ = 2777.5

2777.5(24) = 66660

Each digit will appear an equal number of times in each column. The average value of each column in each addition operation is therefore (1+2+3+4)/4 or 2.5

24 * 2.5 = 60

Adjusting the value for each column gives: (60 * 1000) + (60 * 100) +(60 * 10) + 60

If we consider the number $1234$ , there exists a number that adds $1234$ up to $5555$ , and indeed, that number is $4321$ .

Another example would be: $4231$ + $1324$ = $5555$

This applies for all configuration of these 6-digits numbers; there exists a number that adds that certain number up to $5555$ , and so this gives a "parity" to the 24 numbers as above, and as 24 numbers pairs, then there are: $12$ pairs of numbers that each adds up to $5555$ . Therefore, the sum of the 24 numbers above is:

$5555 \times 12 = \boxed{66660}$

veri nais problem :)

(4+3+2+1)×(1000+100+10+1)×6 is the easiest way I guess.

Take one set of 4 combinations and add together:

1,234 + 2,341 + 3,412 + 4,123 = 11,110

This uses 4 of the 24 combinations 24 / 4 = 6

So the total is 11,110 x 6 = 66,660

As given we can rearrange the columns so that in each column we get the four term sum 11110. There are six such columns which gives 66660

Marvin's solution is very elegant. (I have added the numbers of the 2 rows displayed and got 33330: So the total is the double (66660) since the remaining 2 rows are complementary. According to "You do not need to list out all 24 numbers to find the sum", the solution is correct, but then, too many calculations!)

These numbers are an evenly (well perhaps regularly) spaced comb on the number line. 24 times the average of the extreme values equals 66660

The first number is 1234 and the last is 4321. The average between them is 2777.5; multiply by 24 is 66660.

For each digit, there are 6 ways to choose the other three digits, since all digits are distinct and 3 2 1=6. If you imagine adding the numbers using grade school addition, where you add digits column-wise, for each digit in the sum you are adding every digit from 1 to 4, 6 times each. That means the sum of each column is 6*(1+2+3+4) = 60. So you carry over to each next column. The ones column will be 0 (because rightmost sum is 60), followed by a 6 (60+6 = 66), followed by another 6, and then the leftmost column sums to 66. So, you get

66660

Add smallest (1234) and largest (4321) to give 5555, divide by 2 to get average (2777.5) and x total (24) to give 66660.

Lowest value = 1234 Highest Value = 4321

Sum of these = 5555

5555 / 2 = 2777.5

2777.5 x 24 = 66660