Given how simple this is, I'm not sure if it has been asked before. But I did try searching everywhere and didn't come up with anything. It is based on these two questions I happened to stumble upon:

What is the lowest number you cannot make with a set of 4 numbers and the 4 basic operators? Create all numbers from 0-100 only using all of 1,2,3,4 and 5

My question is similar - using 1, 2, 3, 4, and 5, and only the basic operators (addition, subtraction, multiplication, and division), what is the smallest possible positive integer you cannot make?

The use of brackets is allowed. You must use every single one of the five digits in each case. Unlike the other two questions though, factorials and exponentiation are not allowed, therefore I think this will pose a lot more of a challenge, and make finding the solution more plausible.

Also, just in case it isn't clear, concatenation is also not allowed, and you have to use all the five numbers exactly once for each case. The order of use of the numbers does not matter.

I've tried making it to 100, but got stuck at a certain number which I think might be the solution.

Given how simple this puzzle seems, I apologize in advance if something like this has already been posted.

  • $\begingroup$ Is reordering allowed? Should we use the numbers in that order or we can rearrange them? $\endgroup$ Commented Dec 21, 2017 at 17:35
  • $\begingroup$ Ops! My bad. You can use the numbers in any order. Edited to make that more clear. Thanks for picking that up. $\endgroup$
    – Marcus86
    Commented Dec 21, 2017 at 17:43

2 Answers 2


This cannot be made


The smaller ones can, or so it would seem.

  • $\begingroup$ Yes, that's what I have found, too. $\endgroup$
    – M Oehm
    Commented Dec 21, 2017 at 17:47
  • $\begingroup$ That's where I got stuck on as well! You found it so fast, too. Had you already tried this before and knew it beforehand? Or did you run it through something to figure it out? $\endgroup$
    – Marcus86
    Commented Dec 21, 2017 at 17:58
  • $\begingroup$ The British gameshow ”Countdown” (more recently revived as ”8 out of 10 Cats does Countdown”) not only has the most quick witted ”eye candy” lady of all times, it also has a ”Numbers” section with exactly this kind of problems, so the internet is basically brimming with automated solvers. Or, in so many fewer words, ”I totally cheated” :-) $\endgroup$
    – Bass
    Commented Dec 21, 2017 at 18:24
  • $\begingroup$ I'm not sure that you can say Countdown was revived, given that it's never died (it does hold at least one record for longest running show of its kind, and is still being made and shown 5 times a week, less holidays). $\endgroup$ Commented Nov 29, 2019 at 9:55

The smallest I got that can't be made is


Followed by

79, 86, 92, 94, 97, and 98

I just wrote some c++ to try every combo of numbers and operations!

Here are all the sums https://pastebin.com/Cm3x5gfq


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