8 Operations

Logic Level 1

1 2 3 4 5 6 7 8 9 = 33 \large 1 \; \square \; 2 \; \square \; 3\; \square \; 4\; \square \; 5 \; \square \; 6 \; \square \; 7 \; \square \; 8 \; \square \; 9 = 33

Seven of the eight " \square "s above are filled with " + + ", and the other one with " - ".

Before which integer should the " - " sign be placed to make the equation true?

2 3 4 5 6 7 8 9

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5 solutions

Relevant wiki: Arithmetic Puzzles - Fill in the Blanks

We note that 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 1+2+3+4+5+6+7+8+9 = 45 . If we replace an integer k k between 1 and 9 with 0, the sum of the 9 integers is 45 k 45-k . If we now replace 0 with k -k , the sum of the nine integers is 45 2 k 45-2k . Therefore, 45 2 k = 33 45-2k = 33 , we have k = ( 45 33 ) / 2 = 6 k = (45-33)/2 = \boxed{6} .

Do you think the phrasing is clear? There's a report that they counted the 5th box instead.

Chung Kevin - 4 years, 7 months ago

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My suggestion:

Seven of the eight " \square "s are filled with " + + " and one with " - ". Before which integer should the " - " be placed to make the equation true?

Chew-Seong Cheong - 4 years, 7 months ago
Jason Dyer Staff
Nov 16, 2016

All 9 digits when added makes 45. 45. One needs to swap sign to make it 33 , 33, changing the outcome by 45 33 = 12. 45-33 = 12. This will happen with 12 / 2 = 6. 12 / 2 = 6.

Just wanted to make a de-algebra version, since this problem is solvable by students who haven't had algebra yet.

Jason Dyer Staff - 4 years, 7 months ago

I did the same way.

Sparsh Setia - 4 years, 6 months ago
Jack Ceroni
Nov 4, 2016

To solve this problem, we are given that 7 7 of the 8 8 operation signs in this sequence are addition symbols, and one is a subtraction symbol. First, we must add all of these numbers, which yields the result of 45 45 . We then have to realize that 45 45 is 12 12 greater than the number value which we must obtain, which is 33 33 . Secondly, we can see that to get this result, we will have to subtract one of the integers from this sequence, which we will call y y . If the value of the set of numbers added together, S S is 45 45 , then this will be 45 x 45 - x . When we subtract x x from the sequence however, we are subtracting the number twice, once from the original sequence, and then adding its negative value back into S S . In this context, we are concerned with the magnitudes of the numbers, so we can see this through this:

x + x = x + x = 2 x |-x| + x = x + x = 2x

Since the value of all of the numbers deviates 12 12 units from the value of the hidden operations at a value of 33 33 , we can see that 2 x 2x , which is deviation, is equal to 12 12 . Because of this x = 6 x = 6 . Therefore, the number that we must place the subtraction sign before is 6 6 .

Ondřej Chmelík
May 19, 2019

We need to put minus before 6.

Stacy Blatt
Aug 5, 2017

Again, the sum (45)-2x=33 yields the solution.

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