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This is a riddle my friend gave me and we don't know the answer so we would like some help. The task is to use all of the numbers $0,1,2,...,9$ once each to get a sum of $100$ only using the plus sign.

You can do whatever you want with the numbers so long the plus sign and only that is included. For example, one could concatenate and use the number $12$ or $23$.

Multiplications or exponents or other signs are not allowed.

Bobson Dugnutt
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Themistoklis Haris
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  • That can't be done, if you mean that you can only add $1,2,3,4,5,6,7,8,9$ and only use each once. – Bobson Dugnutt Feb 11 '16 at 21:34
  • @Lovsovs He probably meant with concatenation of the digits, so you can make $89$ with an $8$ and $9$. – Trevor Norton Feb 11 '16 at 21:36
  • Of course just adding is not effective. By combining, one could for example take the number 12 or 43 into account. Maybe there are other more clever ways of combining but I haven't thought of any yet. – Themistoklis Haris Feb 11 '16 at 21:36
  • @TrevorNorton Yeah, that makes more sense. OP, I'd suggest changing the word "combining" with "concatenating". – Bobson Dugnutt Feb 11 '16 at 21:38
  • Certainly that makes more sense, but I'm afraid of discouraging solvers from seeking other possible ways of using the numbers that just concatenations. Maybe I'll go with simply use the numbers. – Themistoklis Haris Feb 11 '16 at 21:40
  • Well...concatenation alone isn't sufficient. if you concatenate two digits, $5$ and $2$ say to make $52$ then the remaining digits sum to $52+45-5-2$ which is divisible by $9$. Maybe if you use a decimal point? – lulu Feb 11 '16 at 21:43
  • I don't know about a decimal point...Perhaps it is allowed but can it lead to a solution? – Themistoklis Haris Feb 11 '16 at 21:44
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    I can't make a solution using decimal points and sums. Doesn't mean it's impossible. I can do it with backslashes (division)! $1 = \frac {148}{296} + \frac {35}{70}$. But that really seems to be far from what you wanted. – lulu Feb 11 '16 at 21:50
  • Can you rotate a plus sign by $45^\circ$ or rotate any of the digits, by $90^\circ$, say? – Steve Kass Feb 11 '16 at 21:59

2 Answers2

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It is impossible, since the sum will always be divisible by $9$. Note that for a number to be divisible by $9$, the sum of its digits must be divisible by $9$; thus, we know that the solution must be divisble by $9$ as $1+\cdots+9=45$ is divisble by $9$, and so it can never be $100$.

To comment on the problem where we can use a decimal point - we have the same problem. Multiply both sides by a power of $10^k$ that makes the solution $x$ integers only - and $10^kx\equiv 1^kx\equiv x\equiv 100\equiv 1\mod 9$. But still the solution multiplied by $10^k$ only adds numbers with digits $1,\cdots,9$ and a number of $0$'s, so $10^kx\equiv 0\mod 9$, which is again a contradiction.
A good website to visit on related problems is cut-the-knot.
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    The best that can be done is $1+2+3+4+5+60+7+8+9=99$. –  Feb 11 '16 at 21:49
  • @YvesDaoust, yes, and there are are many more solutions if we want it to be $99$ –  Feb 11 '16 at 21:50
  • Oh, thanks for sorting out the decimal case (+1). Much appreciated! – lulu Feb 11 '16 at 21:51
  • Well, I've been trying to counterexample this for a while but I have failed so yes I suppose your reasoning is correct and concatenations of numbers or a decimal point cannot lead to a solution. So the point in the puzzle must be a clever way of using the numbers. I just don't know what it is :( – Themistoklis Haris Feb 11 '16 at 21:51
  • @ThemistoklisHaris, we must have another operation to get this to work. –  Feb 11 '16 at 21:53
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I think it is possible if we write $49 + 50+ 1/2 + 38/76$, which sums up to $100$.

YiFan Tey
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Salim ur Rehman
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