34

Having had just checked into the Grand Hotel, Mr. Hilbert slumped into the hotel room armchair with relief. Finally he could have some peace and quiet and solve the quaint riddle his colleague gave him! He pulled a note out from his breastpocket, and on it was simply scribbled:

Fix this equation by adding three mathematical symbols:
$$2 \space\space\space 2 \space = \space9$$ NO letters, numbers, or tampering with the equal sign!

Unfortunately, an hour and thousands of incorrect symbols later, Mr. Hilbert remained hopelessly stuck. Can you help Mr. Hilbert?

McMagister
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Hackiisan
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    How can Mr. Hilbert spend an hour working on a problem in a hotel? Wouldn't he be requested to move to another room once in every few minutes? – JiK Sep 02 '15 at 09:47
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    @JiK The front desk was busy trying to complete Mr Gödel's check-in. – David Tonhofer Sep 02 '15 at 20:11
  • While the Grand Hotel is large, the occupants move slowly =) Summer is also down season for them I suppose? – Hackiisan Sep 03 '15 at 00:34
  • Does 2 + 2 <= 9 qualify as "tampering with the equal sign"? It's still there, just has a < before it. – Darrel Hoffman Sep 03 '15 at 15:40
  • You mean $\leq$? =) – Hackiisan Sep 03 '15 at 18:35
  • Well, yes, but as a programmer, they mean the same thing to me, and I don't have a ≤ on my keyboard. I guess it depends what you define as a "mathematical symbol". (Also, if that worked, I solved it with only 2 symbols...) – Darrel Hoffman Sep 04 '15 at 13:19
  • Also for programmers, 2+2!=9 has two symbols. – Arcturus Jan 08 '16 at 19:54
  • @DavidTonhofer top comment! Hahaha! :P – Mr Pie Apr 08 '19 at 00:28
  • @DarrelHoffman mainly C#, which does incorporate the use of <=. But my friends that also code write it in reverse when we message other, namely, =< to avoid confusion between that and an arrow. I confirmed with one of them today (his name being Joseph) and forgot to reply back in order to apologise for my false information, as I am working on a puzzle at the moment. So sorry about that! You are correct. – Mr Pie Apr 08 '19 at 14:49

12 Answers12

36

I don't know how to do the formatting (thanks McMagister for the edit) but the answer is

$ 2\space\div\space .\overline{2} = 9 $

FriedSaucePots
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    Beautifully solved. – alexmc Sep 02 '15 at 03:09
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    That feel when grade school repeated decimal notation is more foreign to me than the gamma function... – McMagister Sep 02 '15 at 03:12
  • Well done, much faster than Mr. Hilbert! I had hidden a hint in the story text to be revealed later, but clearly it is unnecessary now =) – Hackiisan Sep 02 '15 at 03:15
  • The hint is not so much hidden as I was thinking about it just by reading the title of the question. There is not that many reasons why would one name the usually unnamed (or non-existent) protagonist after a mathematician, besides giving a hint :) – zovits Sep 02 '15 at 09:07
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    For the uninitiated what does the overbar notation mean? I thought that meant complex conjugate? – curious_cat Sep 02 '15 at 12:21
  • @curious_cat repeat the digit forever – Caridorc Sep 02 '15 at 12:34
  • My only issue is that the decimal place isn't really a "symbol", and the overbar isn't really standard mathematical notation (it's common school-level notation, but any real mathematician would write it as the fraction). – Glen O Sep 03 '15 at 04:11
24

$$ 2 \div 2 = .\overline{9} $$

Simply rearranging the symbols used in the intended solution.

GOTO 0
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17

Probably fails the no letter criterion.

$-2+2={d \over dx} 9$

Or using Lagrange notation as a workaround (thanks to McMagister) we can also write

$-2+2= 9'$

Rohcana
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    Use lagrange notation as a work-around? – McMagister Sep 02 '15 at 03:01
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    Yes. While technically you can use a "prime" symbol for differentiation, the intended answer is not a simple "cheat". Otherwise, most of similar type questions can be solved by the trivial $a' + b' = c' + d'$ – Hackiisan Sep 02 '15 at 03:01
14

I came up with this:

$\lceil2\sqrt{2}\rceil = \sqrt{9}$

Hellion
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    implicit multiplcation - getting that fourth operation without a fourth math symbol - love it! – corsiKa Sep 03 '15 at 02:00
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    ...And then losing two math symbols on one math operation (ceil), ending up with one symbol too many... – namey Sep 03 '15 at 21:04
7

My answer:

$$\Gamma(2) + 2 = \sqrt9$$

McMagister
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5

Assuming that perfect formatting is not required:

$ \neg (2 \space 2 = 9) $

That is:

not (twenty-two equals nine)

If we count a pair of parentheses as a single unit, then:

$ \neg (2 + 2 = 9) $

Ypnypn
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5

While the accepted answer was also the first one I thought of, there's also a nice solution with subfactorials:

$$2\;!2 = !\sqrt{9}$$

Peter Taylor
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4

What about :

$2\div2=\#\{9\}$

In case I got the symbols wrong, what I am trying to say is:

two divided by two equals the cardinality of the set of numbers that just contains the number nine

GentlePurpleRain
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Kevin
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  • Nice! While I can't say this is wrong (similar to Anachor's answer), it goes against the spirit of this puzzle by ignoring the value of the number. Otherwise, many other trivial answers exist, such as $(#(2)+2)! = 9$ or $#(2) + 2 = \sqrt{9}$. I would have edited the question to reflect this, but the correct solution has already appeared =P – Hackiisan Sep 02 '15 at 07:15
  • This solution also uses 4 symbols - not the allowed 3. – Eborbob Sep 02 '15 at 12:50
  • @Eborbob Parentheses and brackets are always used a unit and I would consider them to be one symbol. Based on his comments on other answers, Hackiisan seems to agree with me. – Kevin Sep 02 '15 at 13:00
  • @Kevin Fair enough, I'll remember that for future use! – Eborbob Sep 02 '15 at 13:04
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    @Hackiisan $#(2)+2=\sqrt{9}$ isn't trivial, and doesn't ignore the value of any number but the first. One could argue the 'accepted' answer ignores the value of one of the numbers, because any two equal numbers will result in an acceptable result. – corsiKa Sep 02 '15 at 16:42
  • @corsiKa Right, but I don't have to change the current set of numbers to other ones in order to get the correct answer (which of course you may rightfully contest whether $.\bar{2}$ is such a modification). In general, I could define an arbitrary $#(x,y)$ such that $#(a,b) = c$, which defeats the purpose of the puzzle for me subjectively speaking. Similarly, ceil(x), floor(x), not(x), bitand(x,y) are pre-defined discrete many-to-one / one-to-one mappings, which I am also not as fond of. – Hackiisan Sep 03 '15 at 01:03
  • Hey it's your puzzle =) You can have whatever allowed or not allowed. I just notice a lot of puzzles list the particular operations you are allowed to use. This didn't precisely because you're supposed to use ones you don't use very often. I guess what I'm saying is you can't have your cake and eat it too. If you want ones that don't get used much, don't discount the ones that don't get used much. – corsiKa Sep 03 '15 at 01:59
  • Indeed, which is why I upvoted many of them but ticked the intended solution =) People are always more creative than I am! And puzzles like these are often not specific enough to warrant a single answer given the large number of conventions in mathematical notations. – Hackiisan Sep 03 '15 at 18:38
4

How about this?

$\lfloor\sqrt{2}\rfloor + 2 = \sqrt{9}$

Oops, that is 4 symbols. Thanks @corsiKa.

This one uses three symbols:

$2 - 2 = \lfloor9\%\rfloor$

user2023861
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2

How about

$2'\cdot 2' = 9$, where $(\,\cdot\,)'$ denotes the successor function.

Claudius
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  • By that logic, I can always define (⋅)′ to always return 3, irrespective of what is inside. (⋅)′ is not standard notation for the successor function, which means you will have to add letters after these 3 operators to explain what you mean. – CodeNewbie Sep 03 '15 at 13:08
  • I'm pretty sure that $(,\cdot,)'$ is a standard notation for the successor function. At least this is the notation I learned when I was in school. And it is not far fetched to recognise it as such. – Claudius Sep 03 '15 at 13:11
  • Can you provide any links to text where such a notation is used? (Hopefully one that isn't too obscure) – CodeNewbie Sep 03 '15 at 13:16
  • see this question on math.stackexchange: http://math.stackexchange.com/questions/643374/definition-of-successor-function-in-peano-axioms There is a reference to Kleene’s Mathematical Logic that uses this notation. – Claudius Sep 03 '15 at 13:28
  • I stand corrected. Although this is not as common as you expect, indicated by the accepted answer in the above question pointing out that S(n) is a more common notation. – CodeNewbie Sep 03 '15 at 13:31
2

If we can assume these digits are measured we get:

2 + √2 = √9 (true to one significant figure)

Paul Evans
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2

With 2 mathematical operations:

$ 2-2=\{9\} $

HTM
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Artemmm
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