Can you get the numbers 1,2,3,4,5, 6, 7, 8,9 to total 100 just by adding together?

MayCaesar

In questions such as this, I always have to ask: what operations are permitted? Am I allowed to use (.9) meaning 9.999999999..., which technically does not contain a decimal point, as it equals 10? Am I allowed to devise my own operations? Am I allowed to use concatenation, such as (2+3)|9 = 59? Or am I allowed to only put numbers together and use summation?

If it is the latter, then I am not aware of a solution, but depending on the exact problem, there can be one. I have seen one using the (.9) notation, for example, and, I believe there is, at least, one with concatenation.

Dee

No May seriously the question requires no weird mathematical  maneuvering  , one has to think outside the box , think of the total and how you arrive at it 

Happy_Killbot

How much creativity is allowed here?

If you are allowed to "Add a negative" then it's easy: 123 + -45 + -67 + 89 = 100

MayCaesar

@Dee

I wrote a small Python program to run through all possible combinations of expressions, and none of them ended up equal to 100. A LOT ended up equal to 99, 90 or 108, but no 100-s.
And it is pretty obvious why this is the case. Take any possible combination. If you put any two of the free numbers together, the expression increases by a multiple of 9, and if you split any two-digit number into separate digits, the expression decreases by a multiple of 9. And since no number can have more than 2 digits (otherwise it already is bigger than 100), we see that the possible values are 99 +/- multiple of 9.

The way you formulated the problem is somewhat vague, so it is not clear what exactly we are allowed to do. But if all we can do is rearrange the numbers and put a plus between any two numbers, then the solution does not exist.

If I can do some tricks, then I can find many solutions to this problem, for example:
1^{23456789} = 1^{00} (You just write the exponent slightly above the number, so the numbers look as if they were "added together".)

Dee

@Happy_Killbot

The solution involves just as I stated , think of totals and how one arrives at them,  you may use the numbers in pairs , singly or other combos 

Dee

@MayCaesar ;

The solution involves just as I stated , think of totals and how one arrives at them,  you may use the numbers in pairs , singly or other combos .....It’s more lateral thinking than math yet for some strange reason mathematicians get really annoyed at it in 20 years of asking no one had yet solved it 

MayCaesar

@Dee

I am not annoyed when people ask ambiguous questions, which is why I am calmly trying to clarify what it is exactly that you are asking. This is what we, mathematicians, do when the question is not clear to ask: we ask further clarifying questions.

There are no such standard terms in math as "to total", "to get" or "to add together". I do not know what you mean by these terms. I also do not understand what initial data we are dealing with: a set of 9 digits that can be used once each, or a set of 9 digits from which we can infinitely draw digits.
I do not understand what it means to "use the numbers in pairs". "Use" in what sense? And what, again, are "totals"?
Until I understand what it is you are asking, I cannot give you the answer you are looking for.

Dee

@MayCaesar



***I am not annoyed when people ask ambiguous questions,

I never said you were annoyed I was citing the norm when other people get involved 

***/which is why I am calmly trying to clarify what it is exactly that you are asking. This is what we, mathematicians, do when the question is not clear to ask: we ask further clarifying questions.

You’re taking this way too seriously can you not see a bit of lateral thinking may be involved 

***There are no such standard terms in math as "to total", "to get" or "to add together". 

Yet such terms are used and comprehended by many involved in recreational problems like this ,indeed the great Martin Gardener used such terms in his recreational books with math and lateral thinking problems 

****I do not know what you mean by these terms. 

Then ignore the question and move onto something else that appeals 

**** I also do not understand what initial data we are dealing with: a set of 9 digits that can be used once each, or a set of 9 digits from which we can infinitely draw digits.

You use the digits once only , it’s amusing if I ask this to a group of people they get it the only ones who don’t are a certain type of mathematician who cannot use a bit of creativity and imagination 


****I do not understand what it means to "use the numbers in pairs". "Use" in what sense? And what, again, are "totals"?
Until I understand what it is you are asking, I cannot give you the answer you are looking for.

Use in any sense you want ( there’s a huge hint) I cannot help that you do not know that 44 of 32 are a pair to most , also by adding those one gets a total .

Thats fine if you cannot answer 

MayCaesar

@Dee

People who use creativity and imagination explore all possible options. People with narrow thinking only think about one most obvious option, even if the wording of the problem allows for more. They also have trouble communicating their actual questions with others, because they cannot see other people's perspectives.

If I am allowed to think of these terms "in any sense I want", then the solution is in my second comment. I can think of hundreds more solutions, involving various tricks that do not contradict the wording. But there is no way of knowing which of these solutions you classify as the "right" one, as the problem is ill-posed.

Dee

@MayCaesar


**** People who use creativity and imagination explore all possible options. 

Yes 

****People with narrow thinking only think about one most obvious option, 

Yes 

***even if the wording of the problem allows for more.

The wording of the problem is fine , indeed it’s been used as written in several books of lateral thinking problems 

 ***They also have trouble communicating their actual questions with others, because they cannot see other people's perspectives.

I don’t think there’s a problem in communication as otherwise several editors would have pointed it out , maybe it’s an American thing?

****If I am allowed to think of these terms "in any sense I want", then the solution is in my second comment. 

Fine , good for you 

****I can think of hundreds more solutions, involving various tricks that do not contradict the wording. 

Excellent 

****But there is no way of knowing which of these solutions you classify as the "right" one, as the problem is ill-posed.

Yet the great American mathematician Martin Gardener thought it unique but what would he know? 

MayCaesar

@Dee

Martin Gardner worded his problem regarding summation to 100 in a very clear way. Here is his wording:

"An old numerical problem that keeps reappearing in puzzle books as though it had never been analyzed before is the problem of inserting mathematical signs wherever one likes between the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 to make the expression equal 100. The digits must remain in the same sequence. There are many hundreds of solutions, the easiest to find perhaps being

1 + 2 + 3 + 4 + 5 + 6 + 7 + ( 8 x 9 ) = 100

The problem becomes more of a challenge if the mathematical signs are limited to plus and minus. Here again there are many solutions, for example

1 + 2 + 34 – 5 + 67 – 8 + 9 = 100
12 + 3 – 4 + 5 + 67 + 8 + 9 = 100
123 – 4 – 5 – 6 – 7 + 8 – 9 = 100
123 + 4 – 5 + 67 – 89 = 100
123 + 45 – 67 + 8 – 9 = 100
123 – 45 – 67 + 89 = 100

"The last solution is singularly simple," writes the English puzzlist Henry Ernest Dudeney in the answer to Problem No. 94 in his Amusements in Mathematics, "and I do not think it will ever be beaten." 

In view of the popularity of this problem it is surprising that so little effort seems to have been spent on the problem in reverse form. That is, take the digits in descending order, 9 through 1, and form an expression equal to 100 by inserting the smallest possible number of plus and minus signs."

The "unique solution" refers to the only solution involving no more than 4 "+" and "-" signs for the numbers in reverse order:

98 – 76 + 54 + 3 + 21 = 100

You have never stated any of this, which is what I meant when I said that your problem is ill-posed.

Maybe you meant some other similar problem by Gardner, in which case you are welcome to reference the particular section of the particular book in which he stated the problem the exact same way you did.

Dee

@MayCaesar

***Martin Gardner worded his problem regarding summation to 100 in a very clear way. Here is his wording

Correct Martin worded his problem that way , this is not his problem I’ve told you 3 times it involves lateral thinking but no doubt that won’t stop you ignoring again 

****The "unique solution" refers to the only solution involving no more than 4 "+" and "-" signs for the numbers in reverse order:

98 – 76 + 54 + 3 + 21 = 100

That’s his “ unique “ solution to his problem , do read again what I actually said as opposed to what you think I said 

***You  have never stated any of this, which is what I meant when I said that your problem is ill-posed.

It’s not my problem or Martins , funny how he and I understood it perfectly when it was first posed 

***Maybe you meant some other similar problem by Gardner, 

No , it’s still not his 

***In which case you are welcome to reference the particular section of the particular book in which he stated the problem the exact same way you did.

Again your failure to understand what you think I said as opposed to wish I’d said is staggering , maybe that’s something you need to work on?

MayCaesar

@Dee ;

I need to work on many things, but understanding claims worded in an incoherent way is not one of them.

I am asking you once more: state your problem clearly. If you are unable to, then reference any book, article, anything where someone else has stated this problem.
Feel free to also reference anything indicating that Gardner has had any interaction with this problem.

Once I see a well-stated problem, I can give you the solution. Until then, all you are going to receive from me and anyone else trying to solve this problem is a series of clarifying questions.

I have known Martin well enough to say with confidence that he would never work on a problem stated in such a poor way.

Dee

@MayCaesar

**** I need to work on many things, but understanding claims worded in an incoherent way is not one of them.

Yet as I’ve told you repeatedly your poor comprehension skills are your responsibility not mine most everyone else found the problem as stated perfectly reasonable .....I even gave you hints 

*** am asking you once more: state your problem clearly. 

It’s perfectly clear even to a tyro of lateral thinking problems which relegates you from taking  part it seems 

*** If you are unable to, then reference any book, article, anything where someone else has stated this problem.

You accused me several times of  citing a problem by Gardner and you’re such a raging egotist cannot even acknowledge you were wrong despite me telling you otherwise 

****Feel free to also reference anything indicating that Gardner has had any interaction with this problem.

Wow! Thanks for your permission 

****Once I see a well-stated problem, I can give you the solution. 

This from a guy who says stats from Pew Research are not reliable 

****Until then, all you are going to receive from me and anyone else trying to solve this problem is a series of clarifying questions.

You’re wrong again , as 80 per cent of people address the problem without utilizing your insufferable pedantry 

****I have known Martin well enough to say with confidence that he would never work on a problem stated in such a poor way.

Right, of course Martin never used terms , like “pairs” “addition “ or “ totals “ , but you knew him well enough ....Oookay

Happy_Killbot

I think we can conclude that this isn't possible with math, because no matter how you arrange the numbers you can't get 100 without using at least one number twice, subtracting, using decimals or some other trick, because even if you pair up numbers, for example: 12 + 3 + 4 + 56 + 7 + 8 + 9 = 99 is the same as 10 + 3 + 4 + 50 + 6 + 7 + 8 + 9 = 99. This means that the equation we are looking for is: 100 = 1 * (sum of numbers) + 10 * (sum of remaining numbers) This is not possible in this way unless we use numbers 1 - 10 or ignore certain numbers.

All possible numbers adding this way will be divisible by 9 with no remainder.

for example: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45/12 = 5
 12 + 3 + 4 + 65 + 7 + 8 + 9 = 108/9 = 12

Because 100/9 = 11 1/9 There is no solution just by adding.

That all being said, there is a solution that doesn't need math.

I will now accept my Nobel peace prize.

Dee

@Happy_Killbot

Your prize is on the way my friend .....I will let it run for another few days and post you on the solution ....thanks for being a good sport 

MichaelElpers

@Happy_Killbot. That shouldnt be right together because it specifically asked for 100, not 100 with scribbles in it.

Happy_Killbot

@Dee Are you going to post the answer soon? I think this has fizzled out.

Happy_Killbot

@MichaelElpers Stop making fun of me for having bad hand writing!

I don't want to turn into a depressed death drone.

Dee

@Happy_Killbot ;

Yes it has fizzled out because they have seen it before and it fried the lot of them .....I will send it in your messages