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## Impossible Number?

Posted by Karl Sharman on January 6, 2011 – 3:40 pm

Using the digits 1,2,3 & 4 and any of the basic rules of maths – add, subtract, divide, multiply and parentheses, what is the first natural, positive integer that cannot be obtained?

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This post is under “Tom” and has 21 respond so far.

### 21 Responds so far- Add one»

1. 1. Chris Said：

I got stuck at 17.

2. 2. Chris Said：

I’ve made it to 26 so far
1-10: 1,2,3,4,2+3,2+4,3+4,2*4,3(1+2),1+2+3+4
11-16: 3*4-1,3+4,3*4+1,3*4+2,3*(1+4),4(1+3)
17-20: 3(2+4)-1,3(2+4),3(2+4)+1,4(2+3)
21-26: (1+2)(3+4),2(3*4-1),2*3*4-1,2*3*4,2*3*4+1,2(3*4+1)

I’ll post now in view of my previous answer.

3. 3. Chris Said：

27-28: 3(1+2*4),4(2*3+1)

I’m now stuck at 29, but haven’t given up yet.

4. 4. Alex Said：

Ok, not basic maths, but by bringing in a factorial, 29 can be made, 4! = 24
24+3 = 27
27+2 = 29
I’m guessing the use of all numbers isn’t necessary. If it is, then 1*(4!)+3+2

5. 5. Al Gelman Said：

I can get 30 = 2*3(4+1), and 32= 2*4(3+1), and
36 = 3*4(2+1) but I can’t figure out 29,31, or 33,34,or 35

6. 6. Chris Said：

Ditto. I reckon that 29 is the number sought. I’m quite amazed at how many numbers could be found though.

7. 7. SP Said：

29 = 42 – 13

8. 8. Karl Sharman Said：

Good thinking SP!

9. 9. Milly Said：

49?

10. 10. Chris Said：

LOL. Even though it broke the rules, I like SP’s one, especially as it involved 42.

11. 11. Karl Sharman Said：

Chris, I don’t think SP broke the rules?
The question says – “Using the digits…” and asks for the “first positive natural integer” so combining 2 digits to make, say, 12 can be within the parameters of the question…

I have had to work out the answer for myself, and came up with 29, but I think that SP puts a really good slant on it.

I also liked that the question asked for something you cannot do, rather than calculate something you can. It meant I had to break out some paper and a pencil!!

12. 12. Chris Said：

Continuing,
31-40: 31,32,34-1,34,34+1,34+2,34+1+2,41-3,41-2,43-1-2
41-48; 41,42,43,43+1,43+2,34+12,41+2*3,24(3-1)

I’m now stuck at 49.

I could do 41 + 2^3, but if that’s allowed, I’m stopping.

13. 13. Curtis Said：

Then 31 would be 43-12

14. 14. Chris Said：

Hi Curtis. 31 is 31 is simpler. I didn’t realise that string catenation was to be regarded as a mathematical operation until post 8 from Karl.

15. 15. Karl Sharman Said：

Non- string catenation – 29 – Chris
String Catenation – 49 – Milly

16. 16. Jake Said：

0

17. 17. Jake Said：

never mind, u put natural, if i remember right that means it has to be greater than 0. i skipped over that part.

18. 18. DP Said：

besides, 0 = 3 – 2 – 1

19. 19. jakob Said：

i get 1

20. 20. PIanissimo Said：

What about extending the question in the opposite direction? What is the smallest whole number that cannot be calculated this way. It is less than zero.

21. 21. DW516 Said：

I see that this is an old post, but just wanted to let you know that there does not exist a number that cannot be computed with the given constraints being that you are allowing the natural number addition group with 2 relatively prime numbers. Hell, if you allowed just 1 alone you get all the numbers by addition. I am just wondering how every one decided that 29 is the first one that can not be computed. Try 1+1+1+(total of 29 times)+1 and your done. More interesting would be a semi-group without unity and non-relatively prime members.

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