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## Equals 1.

Posted by Karl Sharman on July 5, 2011 – 1:41 pm

In how many ways can you use all the digits 1,2…9 and the operands + and – to create the equation 1?2?3?4?5?6?7?8?9=1

You can put the numbers in any order if it makes you happy, all 9 digits must be used, because I insist, and you can only use add and subtract . Leading minuses – oh, alright then.

Oh, and if I haven’t already mentioned it, they must equal to 1.

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This post is under “Tom” and has 18 respond so far.

### 18 Responds so far- Add one»

1. 1. slavy Said：

Hi Karl, nice problem I like it Just have a question though – can we glue digits together or we should put a sign between every two of them?

2. 2. Karl Sharman Said：

Sorry Slavy, in this one you can’t ‘glue’ them together.

In the 1 equals question you can, although I haven’t explicitly said that in the question…..

3. 3. Knightmare Said：

method:the force

4. 4. Karl Sharman Said：

The force is strong with this one, but the midichlorian count appears to be too low…..

5. 5. Viji Said：

-1+2+3+4-5+6-7+8-9 = 1

6. 6. Jan Said：

I found 23 and im confident i found them all.

7. 7. Karl Sharman Said：

Jan’s confidence is ill judged – I have more than that… Join forces with the Knightmare of the Old Republic see if you can help each other….

8. 8. Jan Said：

hmmm, my computer seems to agree with me. show me one that i dont have.

+1+2+3+4+5-6-7+8-9
+1+2+3+4-5+6+7-8-9
+1+2+3-4-5-6-7+8+9
+1+2-3+4-5-6+7-8+9
+1+2-3-4+5+6-7-8+9
+1+2-3-4+5-6+7+8-9
+1-2+3+4-5+6-7-8+9
+1-2+3+4-5-6+7+8-9
+1-2+3-4+5+6-7+8-9
+1-2-3+4+5+6+7-8-9
+1-2-3-4+5-6-7+8+9
+1-2-3-4-5+6+7-8+9
-1+2+3+4+5-6-7-8+9
-1+2+3+4-5+6-7+8-9
-1+2+3-4+5+6+7-8-9
-1+2-3+4-5-6-7+8+9
-1+2-3-4+5-6+7-8+9
-1+2-3-4-5+6+7+8-9
-1-2+3+4-5-6+7-8+9
-1-2+3-4+5+6-7-8+9
-1-2+3-4+5-6+7+8-9
-1-2-3+4+5+6-7+8-9
-1-2-3-4-5+6-7+8+9

9. 9. Jan Said：

are there more or what?
if there is no glueing or other type of funny stuff this has to be the right answer.

10. 10. Wizard of Oz Said：

Without using a computer I too can only find 23.

If we collect all the digits that sum to 22, then the rest sum to 23. If the former collection is made negative and the latter positive then we have a sum of 1, as required.

So the exercise comes down to how many subsets of digits in the superset 1,2,3 . . . 9 can we find that sum to 22.

Like Jan I can only find 23:

985 9841 9832
976 9751 9742 97312
9652 9643 96412
95431

8761 8752 8743 87421
8653 86521 86431
85432

7654 76531 76432
75431

11. 11. cazayoux Said：

I’m with Jan, exactly 23.

12. 12. Karl Sharman Said：

Here’s my answer(s) – I’ve got 24, now I am going to go through them and check for duplicates or errors! Or whether I can count to 24!

1+2+3+4-5+6+7-8-9 = 1
-1+2+3-4+5+6+7-8-9 = 1
1-2-3+4+5+6+7-8-9 = 1
1+2+3+4+5-6-7+8-9 = 1
-1+2+3+4-5+6-7+8-9 = 1
1-2+3-4+5+6-7+8-9 = 1
-1-2-3+4+5+6-7+8-9 = 1
1-2+3+4-5-6+7+8-9 = 1
1+2-3-4+5-6+7+8-9 = 1
-1-2+3-4+5-6+7+8-9 = 1
-1+2-3-4-5+6+7+8-9 = 1
-1+2+3+4+5-6-7-8+9 = 1
1-2+3+4-5+6-7-8+9 = 1
1+2-3-4+5+6-7-8+9 = 1
-1-2+3-4+5+6-7-8+9 = 1
1+2-3+4-5-6+7-8+9 = 1
-1-2+3+4-5-6+7-8+9 = 1
-1+2-3-4+5-6+7-8+9 = 1
1-2-3-4-5+6+7-8+9 = 1
1+2+3-4-5-6-7+8+9 = 1
-1+2-3+4-5-6-7+8+9 = 1
1-2-3-4+5-6-7+8+9 = 1
-1-2-3-4-5+6-7+8+9 = 1
-1-2-3+4+5+6-7+8-9 = 1

NB I haven’t put any other effort (than sorting your calculations) into this problem.

13. 13. Chris Said：

Hi Karl. Your 7th and 24th line are duplicates

I’ve sorted the lines using the signs

-1-2-3+4+5+6-7+8-9 = 1
-1-2-3-4-5+6-7+8+9 = 1
-1-2+3-4+5-6+7+8-9 = 1
-1-2+3-4+5+6-7-8+9 = 1
-1-2+3+4-5-6+7-8+9 = 1
-1+2-3-4-5+6+7+8-9 = 1
-1+2-3-4+5-6+7-8+9 = 1
-1+2-3+4-5-6-7+8+9 = 1
-1+2+3-4+5+6+7-8-9 = 1
-1+2+3+4-5+6-7+8-9 = 1
-1+2+3+4+5-6-7-8+9 = 1
1-2-3-4+5-6-7+8+9 = 1
1-2-3-4-5+6+7-8+9 = 1
1-2-3+4+5+6+7-8-9 = 1
1-2+3-4+5+6-7+8-9 = 1
1-2+3+4-5-6+7+8-9 = 1
1-2+3+4-5+6-7-8+9 = 1
1+2-3-4+5-6+7+8-9 = 1
1+2-3-4+5+6-7-8+9 = 1
1+2-3+4-5-6+7-8+9 = 1
1+2+3-4-5-6-7+8+9 = 1
1+2+3+4+5-6-7+8-9 = 1
1+2+3+4-5+6+7-8-9 = 1

14. 14. Karl Sharman Said：

AAAAAAAAARRRRRRGGGGGGGHHHHH! Had to be the last line too! Thanks Chris!

15. 15. Chris Said：

My pleasure

16. 16. Karl Sharman Said：

Someone has recently pointed out that to get some extra answers you could put a + in front of the 1’s as well as just the one’s. Seems redundant, but on paper, regardless of the remarkable similarities between 1 and +1, gives an extra 12!!

17. 17. swagat Said：

2-1+3+4+5-6-7-8+9

18. 18. Chris Said：

Hi swagat. I was going to say that the 1 must come before the 2. But I see Karl has said any order is allowed. So there are 23 * 9! = 8346240 solutions. I’ll post them all later (NOT).

If leading +’s are allowed then that’s becomes 23 * 9! * 212 = 34186199040 solutions.

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