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April 6th, 2018, 10:51 AM   #1
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Question How to fill out the right numbers in a problem which seems guessing?

First off, I'm not good at guessing and probably this is the most scary kind problems I often see in exams. Is there any way to solve this situation without "guessing" or just plugin numbers randomly?

The problem is as follows:

In each circle from the figure below write down an integer number between $1$ to $9$. Each one of them must be different from each other. The resulting sum from those numbers joined by three circles in a straight line and those joined by arrows in the direction indicated be equal to $18$. Which number must be written in the circle painted by an orange shade? (See attachment for the figure)

$\textrm{The alternatives given were:}$
  • 7
  • 4
  • 5
  • 3
  • 6

The only thing I could come up with was to make a sequence of dividing $18$ into different choices for summing that number between $1$ to $9$.

Therefore:

$1+8+9=18$

$2+7+9=18$

$3+6+9=18$

$4+5+9=18$

and the rest would be repeating those numbers in different order like

$5+4+9=18$

$6+3+9=18$

$7+2+9=18$

$8+1+9=18$

So the preceding can't be:

But this other combination can also reach to $18$:

$8+7+3=18$

$8+6+4=18$

However at this point I got tangled up with many choices therefore end up with no clear clue to take to get $18$.

How can I solve this problem without just guessing?. Is there an algorithm, formula or method that can be followed orderly to avoid wasting much time?.

I would like someone could re drawn the question and explain me with much details possible.

Please do not just fill out the drawing right away. I know maybe you have the ability to do this, but this is not what I'm looking for but rather a more step-by-step solution which has proven that it works in many similar scenarios.
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April 6th, 2018, 11:51 AM   #2
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The top chain pointing SE has only 2 circles but this would require both to be equal to 9. Am I missing something?
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April 6th, 2018, 12:33 PM   #3
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That chain of two circles is a mistake in the diagram.

In general, this type of problem can be difficult. In this case, though, some simple algebra suffices.

Let x denote the required number.
Let the two numbers nearest the lower right corner of the image total y.
x + y = 9 (because the remaining numbers complete two rows that total 18 each, and all nine numbers total 45).

If z denotes the total of the four numbers other than x in the two rows containing x,
2x + z = 18 + 18, but y + z = 18 + 18, so y = 2x.

It follows that x + 2x = 9, and so x = 3.
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April 6th, 2018, 01:09 PM   #4
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Sorry as it was mentioned there was an error in the figure. I have corrected it as it was shown in my book. I had to redrawn it so it was my mistake. So as it is now can it be solved?.
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April 6th, 2018, 01:33 PM   #5
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Quote:
Originally Posted by skipjack View Post
That chain of two circles is a mistake in the diagram.

In general, this type of problem can be difficult. In this case, though, some simple algebra suffices.

Let x denote the required number.
Let the two numbers nearest the lower right corner of the image total y.
x + y = 9 (because the remaining numbers complete two rows that total 18 each, and all nine numbers total 45).

If z denotes the total of the four numbers other than x in the two rows containing x,
2x + z = 18 + 18, but y + z = 18 + 18, so y = 2x.

It follows that x + 2x = 9, and so x = 3.
Can we rewind a bit?. I think for someone like me it is difficult to picture what you're trying to say without a visual aid. Which are the circles you're referring to as the ones on the lower right corner?

Whatever you say after that I'm totally lost. i.e why $18+18$?.

If I follow the route of using algebra maybe I can end up with several equations and well I don't think this is something which can be solved quickly.

Maybe can you explain with more details?.
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April 6th, 2018, 02:00 PM   #6
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I meant the lower right corner of the rectangular thumbnail. The 18 + 18 refers to the sum of the numbers in two of the rows that are given as totalling 18 each. Just consider one line at a time of my post carefully to understand what I did.
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April 7th, 2018, 05:51 AM   #7
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Quote:
Originally Posted by skipjack View Post
I meant the lower right corner of the rectangular thumbnail. The 18 + 18 refers to the sum of the numbers in two of the rows that are given as totalling 18 each. Just consider one line at a time of my post carefully to understand what I did.
Maybe I'm too dumb for these things but I'm still not getting it. Perhaps is there any other way?

There are also other zig zag lines which seem to confuse me as well, from your words does it mean that it is unnecessary to fill out the remaining circles to solve this problem?
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April 7th, 2018, 10:55 AM   #8
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You don't need to fill out the remaining circles. Only straight rows of three circles are considered initially, so ignore the zigzag lines. Choose two parallel lines of 3 circles each. Their numbers total 18 + 18. What do the numbers in the remaining 3 circles total?
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April 10th, 2018, 12:19 PM   #9
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You don't need to fill out the remaining circles. Only straight rows of three circles are considered initially, so ignore the zigzag lines. Choose two parallel lines of 3 circles each. Their numbers total 18 + 18. What do the numbers in the remaining 3 circles total?
There is a very important caveat or warning which should had been put in bold letters for that any method to be used be held true and that is. You cannot allow turning and no more than three circles can sum $18$. If this thing was added to the question then everything what you said makes sense. Or at least this is what I concluded. I'm not sure if it sounds obvious or anything but if you read the question and solely focus in the part "follow the arrows" there are multiple interpretations of that specific part in the problem which can led to misleading results and more importantly not to solve the problem.

Okay returning to your method (adding this caveat).

The remaining would be $9$ to that added with $36$ becomes $45$ but I don't know where to go from there. Maybe can you reuse my drawing as a visual aid to know what you're trying to say?. Because I'm still lost at this problem.
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April 11th, 2018, 02:45 AM   #10
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Those three remaining circles that total 9 are the orange shaded circle, whose number I called x, and two other circles, whose numbers have a total I called y. Hence x + y = 9.

There are two arrowed rows of three (whose numbers therefore total 18 in each row) that include the orange shaded circle. These two rows contain the orange shaded circle twice and four other circles. I used z for the total of the numbers in those four other circles, so that 2x + z = 18 + 18.

The four circles that contributed towards z and the two circles that contributed towards y form, when considered together, two arrowed rows of three circles, so their numbers total 18 + 18, which means that y + z = 18 + 18. As 2x + z = 18 + 18 is already known, it follows that y = 2x. As x + y = 9, it now follows that x + 2x = 9, which implies that x = 3.
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