Solve the -1-2-3-4-5-6-7-8-9-10=11 Math Problem with Brackets

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Discussion Overview

The discussion revolves around the mathematical problem of inserting brackets into the expression -1-2-3-4-5-6-7-8-9-10 to make it equal to 11. Participants explore various solutions, the number of possible configurations, and the implications of including or excluding multiplication.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • Some participants propose that there are numerous ways to insert brackets, with one participant claiming to have found 13,639 solutions using a program.
  • Others argue that the problem may have been intended to be solved with only one set of parentheses, suggesting that this leads to a unique solution.
  • A participant mentions finding 551 solutions without using multiplication, while another states that there are 17 fundamentally different solutions when not allowing for nested brackets.
  • Some participants discuss the potential for even more solutions if multiplication is permitted, with estimates suggesting a significantly larger number of configurations.
  • There is a mention of the equivalence of certain solutions based on the order of operations and writing style, indicating that different representations may yield the same mathematical outcome.
  • One participant provides a detailed list of 17 distinct solutions, emphasizing that all solutions can be reduced to these forms.
  • Concerns are raised about the classification of solutions and how nested brackets can be reduced to simpler forms.

Areas of Agreement / Disagreement

Participants generally agree that there are many solutions to the problem, but there is no consensus on the exact number or the implications of including multiplication. The discussion remains unresolved regarding the best approach to finding solutions and the interpretation of the problem's requirements.

Contextual Notes

There are limitations regarding the assumptions made about the use of multiplication and the definition of distinct solutions. Some solutions may be considered duplicates based on different representations, and the discussion includes unresolved mathematical steps regarding the total count of solutions.

zoki85
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I saw it in local newspapers:

-1-2-3-4-5-6-7-8-9-10=11

Insert brackets on the left side to make the sum correct.
I solved it like -1-(2-3)-4-(5-6-7-8-9)-10=11 but there are more
possibilities.How many solutions can you find?
 
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zoki85 said:
-1-2-3-4-5-6-7-8-9-10=11

Insert brackets on the left side to make the sum correct.
I solved it like -1-(2-3)-4-(5-6-7-8-9)-10=11 but there are more
possibilities.How many solutions can you find?

Too many to list. Keeping in mind that "(2-3)(-4-5)-6" implies a multiplication, and that you can subdivide the number 10 by something like "-9-1(0)" or "(-9-1)(0)", there's worlds of possibilities! I wrote a program that found 13,639 solutions using one, two, or three pairs of parentheses.

However! I think the writers of the problem were after something a little more sneaky. Try solving the problem with only ONE set of parentheses, and you'll notice that there's only ONE solution!

DaveE
 
13639 is really astronomical number.And the number of possibilities could be even much larger than that.I'm not sure if they predicted multiplication as legal option.How many possibilities do you find without multiplication usage?
 
zoki85 said:
13639 is really astronomical number.And the number of possibilities could be even much larger than that.I'm not sure if they predicted multiplication as legal option.How many possibilities do you find without multiplication usage?

Without multiplication, my program found 551 solutions. But a bunch of those are duplicates. For example, the program considers the following answers to be different, when really, they're the same:

-1-2-3-4-(5-6)-(7-8-9-10) = 11
-1-2-3-4-((5-6))-(7-8-9-10) = 11
(-1)-2-3-4-(5-6)-(7-8-9-10) = 11

But, there's still lots of solutions. Here's a few it found:

-1-2-3-4-(5-6)-(7-8-9-10) = 11
-1-2-3-4-(5-(6-(7-8))-9-10) = 11
-1-2-3-4-(5-(6-(7-8-9-10))) = 11
-1-(2-(3-4-(5-6))-7-8-9)-10 = 11
-1-(2-(3-4-5)-6-7-8-(9-10)) = 11
-1-((2-3-4-5)-6-7-8)-9-10 = 11
-1-(2-3)-4-(5-6-7-8-9)-10 = 11

There's tons, though. I could go on.

The point is that there's only ONE solution if you just use one pair of parentheses. I think THAT's the challenge they were offering.

My program found:
1 pair - 1 solution
2 pair - 32 solutions
3 pair - 518 solutions

So, see if you can find the one solution that only uses a single pair of parentheses!

DaveE

[edit]
So, modifying my program to account for *most* (but not all) of the duplicates, and not allowing for multiplication, it finds:
1 pair - 1 solution
2 pair - 20 solutions
3 pair - 189 solutions
[/edit]
 
Last edited:
If we don't allow multiplications , there
are only 17 basicaly different solutions without nested brackets:

1: -(1-2-3-4-5)-6-7-(8-9-10) =11
2: -(1-2-3-4)-(5-6)-(7-8)-(9-10)=11
3:-(1-2-3-4)-5-(6-7-8-9)-10 =11
4:-(1-2-3)-(4-5-6-7)-8-(9-10) =11
5: -(1-2-3)-(4-5-6)-(7-8-9)-10 =11
6: -(1-2)-(3-4-5-6-7)-(8-9)-10=11
7: -(1-2)-(3-4)-5-6-(7-8-9-10)=11
8:-(1-2)-3-(4-5)-(6-7)-(8-9-10)=11
9: -(1-2)-3-4-(5-6-7-8)-(9-10) =11
10:-1-(2-3-4-5-6-7-8)-9-10 =11
11:-1-(2-3-4)-5-(6-7)-(8-9-10) =11
12: -1-(2-3)-(4-5-6)-7-(8-9-10)=11
13:-1-(2-3)-(4-5)-(6-7-8)-(9-10)=11
14:-1-(2-3)-4-(5-6-7-8-9)-10 =11
15: -1-2-(3-4-5-6)-(7-8)-(9-10) =11
16: -1-2-(3-4-5)-(6-7-8-9)-10 =11
17: -1-2-3-4-(5-6)-(7-8-9-10) =11


I obtained them by program written in Turbo Pascal

Code: 
PROGRAM Brackets (INPUT,OUTPUT);

VAR
   i,j,counter,sum:INTEGER;
   t : ARRAY[1..10] OF INTEGER;
   open:BOOLEAN;
BEGIN

counter:=0;
t[1]:=1;
FOR i:=2 TO 10 DO t[i]:=2*t[i-1];
FOR i:=512 TO 1023 DO
BEGIN
sum:=0;
FOR j:=1 TO 10 DO
   IF t[11-j] AND i:=0 THEN sum:=sum+j
                       ELSE sum:=sum-j;
   IF sum=11 THEN
BEGIN
   INC(counter);
   WRITE(counter:5,':');
FOR j:=1 TO 10 DO
   IF t[11-j] AND i=0 THEN WRITE('+',j)
                      ELSE WRITE('-',j);
WRITELN('=11');
END;
END;
END.

The program is written to be short and quick so maybe isn't
obvious at first glance how works.
The resulting listing given above corresponds to program output when we get
rid of the brackets on the left side of the expressions.
One feature of this program is that it can be modifyed to
find all possibilities with and without nested brackets.
There are 7628 of these in total!No more no less.
However all of the solutions with nested brackets can be reduced
to the 17 basicaly different ones given above.
For example:

-1-2-3-4-(5-(6-(7-8-9))-10)=11
-1-2-3-4-(5-(6-(7-8-9-10)))=11

are reducible to the solution #17.

EDIT:
I didn't bother myself to compose the program that will account
for multiplication options as well.
But my rough estimate gives me an upper bound
~[itex]7628\sqrt{20}[/itex] for number of all the solutions with these options.
 
Last edited:
tehno said:
However all of the solutions with nested brackets can be reduced
to the 17 basicaly different ones given above.

Effectively, yes, there's a limited amount of +'s and -'s that you can apply to each term. So essentially, by adding parentheses, you're granting the ability to make some of the numbers positive, or leave them negative. The number 1 always has to be negative, but anything else can be switched one way or another. So, in theory, there are 2^9 (512) possibilities, and (I trust) only 17 of those are valid solutions.

But what that means is that a solution like this:

-1-(2-(3-4-5)-6-7-8-(9-10)) = 11

Would be equivalent to your solution #14:

-1-(2-3)-4-(5-6-7-8-9)-10 = 11

Now, that's sort of true, but I'd argue that those really are pretty distinct in terms of order of operations and in writing style.

DaveE

[edit]
To be a bit more clear, the 17 solutions are:

-1,+2,+3,+4,+5,-6,-7,-8,+9,+10
-1,+2,+3,+4,-5,+6,-7,+8,-9,+10
-1,+2,+3,+4,-5,-6,+7,+8,+9,-10
-1,+2,+3,-4,+5,+6,+7,-8,-9,+10
-1,+2,+3,-4,+5,+6,-7,+8,+9,-10
-1,+2,-3,+4,+5,+6,+7,-8,+9,-10
-1,+2,-3,+4,-5,-6,-7,+8,+9,+10
-1,+2,-3,-4,+5,-6,+7,-8,+9,+10
-1,+2,-3,-4,-5,+6,+7,+8,-9,+10
-1,-2,+3,+4,+5,+6,+7,+8,-9,-10
-1,-2,+3,+4,-5,-6,+7,-8,+9,+10
-1,-2,+3,-4,+5,+6,-7,-8,+9,+10
-1,-2,+3,-4,+5,-6,+7,+8,-9,+10
-1,-2,+3,-4,-5,+6,+7,+8,+9,-10
-1,-2,-3,+4,+5,+6,-7,+8,-9,+10
-1,-2,-3,+4,+5,-6,+7,+8,+9,-10
-1,-2,-3,-4,-5,+6,-7,+8,+9,+10

And there are multiple ways of writing each of these with parentheses, as demonstrated above.
[/edit]
 
Last edited:
Classification of reduction depends on how one uses the no-nested brackets code for composing the expanded code for nested brackets solution.
Point is there are 7628 solutions and that all are reducible to 17 solutions.

cheers.
 
That's absolutely insane!More than 7628 solutions if multiplication can be used...Ok,I'll trust You guys.
Does that mean that they expect to find the best solution,like that under
number #10 in tehno's post (with only 1 pair of brackets)?
By the way ,how many solutions exists with unnested brackets but with allowed multiplication ?
 

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