Using the digits 0,1,2,3,4,5,6,7,8,9 only once make the sum of 2 five-digit numbers as large as possible?

Question

Then using the numbers 0,1,2,3,4,5,6,7,8,9 only once make the difference the smallest possible positive interger? The sum seems like it should be 97531+86420=183,951 but I am not sure. The difference seems like it should be 97531-86420=11,111 but I am not sure. Are these correct solutions.

Answer 1

For the sum, you are on the right track... Just keep using the biggest numbers to the left.

97531 + 86420 = 183,951.

Note, that you could switch the digits in the same place and still get the same answer, so there other combinations that would give the same sum. (e.g. 96520 + 87431 = 183,951).

For the smallest difference, if you could get negative numbers you'd want a small number minus a big number with something like: 10234 - 98765 = -88,531

However, in rereading your question I noticed it said "smallest positive integer"... I think you can do better than what you picked...

You want the first digits to have a minimal positive difference of 1, but the other digits you want to have negative results. On the bottom you want to be subtracting n9876. On top you want to have (n+1)0123. That leaves 4 and 5 for n and n+1.

The final difference is: 50123 - 49876 = 247

Answer 2

In a five-digit number, the first digit is worth 10,000 and thus should be the largest possible. Use both 8 and 9 to get a total of 17.

The second digit is worth 1,000 and thus is less important than the first digit but more than all the others. Use 6 and 7.

Continue for the remaining positions.

97531 and 86420 is a possible combination, but 87531 and 96420 is also perfectly all right.

The sum will always be 183,951 no matter which digit goes in which of the two numbers (as long as the order is correct).

This is the solution for the sum which should be as large as possible.


I think you should look for the smallest difference for the same numbers.

The difference between 96420 and 87531 is smaller than 11,111 so that answer is incorrect I'm afraid.

Here order is important. The difference has to be a positive integer, so you have to start with {9,8}. But thereafter it is best to subtract as much as possible. This means the higher digit should go in the second number. The digits should be {6,7}, then {4,5}, then {2,3}, then {0,1}.

This results in numbers: 96420 and 87531

Try swapping 6 and 7. The sum stays the same, the difference increases.
Try swapping 4 and 5. And so on.

Answer 3

For the largest possible sum, yes you need to put the largest digits as far to the left of the numbers as you can. Any combination that will put the 9 and 8 as 1st digit, 7 and 6 as 2 digit, etc. will give the same result, for example
96421+87530=183,951.

As for the smallest difference (assuming it has to be positive),
The smallest difference between the first digits is 1.
To minimize the result you need to make the last 4 digits as small as possible for the first number (x0123), and as large as possible for the second number (x9876).
This leaves you with 4 and 5 for the leading digits:
50123 - 49876 = 247.

Answer 4

23