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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?

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


  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:

    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

    Any advances?

  16. 16. Jake Said:


  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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