# All digits with one operation

I can't find this very easy, simple puzzle here, so here goes:

Using all ten digits from 0 to 9 exactly one time, find an equation (using =, not !=, don't be silly) that uses only one of the operators [+, -, /, *].

To be clear, I'm looking for answers of the form

abcd + efg = hij

ab * cd = efghij

etc.

I found an answer for addition/substraction years ago as an elementary school homework problem. But I'm curious if there's one for multiplication/division.

• For just abc + def = ghij you can go up to 96 examples. – oleslaw Oct 26 '16 at 10:14
• To use only one of the operators mean one instance of the operator or one type of operators (i.e. a+b+c=d uses only the '+')? – oleslaw Oct 26 '16 at 10:41
• Only one instance, as per the answers below – rarpm Oct 26 '16 at 10:44

There are lots of examples for addition:

There are 84 examples if we agree that a+b=c is no different than b+a=c. All of them are in form of "abc + def = ghij" and "ab + cdef = ghij".
i.e. 246 + 789 = 1035

The same lots of substraction examples (switch the numbers from addition):

i.e. 1035 - 789 = 246

Many multiplication examples:

There are 22 examples of multiplication. They appear in the forms of "a x bcde = fghij" and "ab x cde = fghij"
i.e. 27 * 594 = 16038

The same many division examples (switch the numbers from multiplication):

i.e. 16038 / 594 = 27

Some things to take into consideration when finding such examples:

1. "1" is usually in front of the largest number you create (statistics).
2. You have to decide how many digits to "use up" for each number you create.
3. The last digit is the easiest to find because it is not dependent on any other digit.
The rest just lines up after you do this.

• This is the most complete answer. As a comment, I don't think there are "lots", maybe a dozen or so. – rarpm Oct 26 '16 at 10:51
• I would only count 48 examples of addition/substraction ( a+b = b +a). There's only 9 different examples of multiplication/substraction! – rarpm Oct 26 '16 at 12:22
• In fact, there's only 9 combinations for multiplication/division, apart from the 48 for addition/substraction – rarpm Oct 26 '16 at 12:33
• @rarpm A little more than that :) – oleslaw Oct 26 '16 at 12:46
• Oh yeah! I didn't check for the single-digit solutions :P – rarpm Oct 26 '16 at 14:00

There are multiple solutions: Here are two I quickly found:

  742
+356
----
1098

.

  724
+365
----
1089

Reasoning:

$7+3 = 10$, $4 + 5 = 9$, $2+6 = 8$.
You can arrange them in every combination so that 7 and 3 go on the hundreds places, and the 2 other combinations can be on the tenths and units places because there is no carriage from their sum. You can re-arrange them to get 8 combinations I think.

Working on the multiplication.

• Yes, these are I found as well! And you can of course invert the digits, too. – rarpm Oct 26 '16 at 10:29

Many examples can be built for additions and subtractions.

Here is my multiplication example -

$5694 * 3 = 17082$ which can also be presented as $17082 / 3 = 5694$

• Nice! May I ask how you found it? – rarpm Oct 26 '16 at 10:29
• Microsoft Excel! :) – Techidiot Oct 26 '16 at 10:31

I brute-forced a list of 2548 equations meeting these criteria. There are in total 5096 equations with the operator on either side of the equal sign.