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April 6th, 2018, 10:51 AM  #1 
Member Joined: Jun 2017 From: Lima, Peru Posts: 40 Thanks: 1 Math Focus: Calculus  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) $\textrm{The alternatives given were:}$
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 stepbystep solution which has proven that it works in many similar scenarios. 
April 6th, 2018, 11:51 AM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 351 Thanks: 192 Math Focus: Dynamical systems, analytic function theory, numerics 
The top chain pointing SE has only 2 circles but this would require both to be equal to 9. Am I missing something?

April 6th, 2018, 12:33 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 18,838 Thanks: 1564 
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. 
April 6th, 2018, 01:09 PM  #4 
Member Joined: Jun 2017 From: Lima, Peru Posts: 40 Thanks: 1 Math Focus: Calculus 
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?. 
April 6th, 2018, 01:33 PM  #5  
Member Joined: Jun 2017 From: Lima, Peru Posts: 40 Thanks: 1 Math Focus: Calculus  Quote:
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?.  
April 6th, 2018, 02:00 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,838 Thanks: 1564 
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.

April 7th, 2018, 05:51 AM  #7  
Member Joined: Jun 2017 From: Lima, Peru Posts: 40 Thanks: 1 Math Focus: Calculus  Quote:
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?  
April 7th, 2018, 10:55 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 18,838 Thanks: 1564 
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?

April 10th, 2018, 12:19 PM  #9  
Member Joined: Jun 2017 From: Lima, Peru Posts: 40 Thanks: 1 Math Focus: Calculus  Quote:
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.  
April 11th, 2018, 02:45 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 18,838 Thanks: 1564 
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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