1
          /\
         /  \
      5 / 12 \ 6
       /______\
           4       

          /\
         /  \
      6 / 21 \ 7
       /______\
           5      

          /\
         /  \
      4 / ?  \ 8
       /______\
           10

The question is to find the missing term.

Answers - a) 22 b) 30 c) 32 d) None of the above

Solution given in the book -

Clearly we have: (5X6X4)/10 = 12

              (6X7X5)/10 = 21

              (4X8X10)/10 = 32

Hence the answer is 32

My confusion in this solution - Why they came up with 10 (in bold)? They could come up with any other numbers for example 5 or 20 or anything else. Why 10?

My solution -

Clearly we have (5X4 - 6) - 2 = 12

            (6X5 - 7) - 2 = 21

            (10X4 - 8) - 2 = 30

So my answer is 30

My question - I didn't find anything wrong with my answer. So why will I take my book's solution only? But surprisingly with both approach the answers are not the same. My answer is 30 and the solution book gave has 32 as an answer. Both 30 and 32 is among the choice of answers in the question.?

Man_From_India
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    Excellent example that these things don't necessarily have a logic equally reachable by all...and, in fact, many of these games are rather nonsensical. Your solution looks to me perfectly valid and, in fact, even wittier than the book's. – DonAntonio Apr 22 '14 at 02:45
  • The claim would be that your solutions are more complicated than the book's. If your multiplies didn't involve the same two sides of the triangle, I would agree. The $-2$ supports the claim. As it is, I think the claim is questionable. Often, what is simplest or more reasonable is in the eye of the beholder. – Ross Millikan Apr 22 '14 at 02:55
  • But in examination it's hard to come up with different approaches, especially when the time is a constraint. So if I come up with the solution I gave, instead of the one the book gave I will lead to a different answer. And as a result I will end up choosing a wrong number as an answer. What to do, then? – Man_From_India Apr 22 '14 at 03:08
  • @RossMillikan I don't think I got what you said. Please tell me more. – Man_From_India Apr 22 '14 at 03:10
  • These are more feeling questions than math. If the center number on the second were $35$, you could claim that $(6 \times 7 -5)-2=35$. I would then find your claim less credible because it doesn't use the same sides of the triangle as the first, so you don't have a unique solution to the third. The book's answer is more uniform than yours, in that it multiplies all the exterior numbers. On the other hand, it pulls the divisor $10$ out of thin air, as you do with the $-2$. I would accept the book answer as better, but not by much. Others may disagree. – Ross Millikan Apr 22 '14 at 03:45
  • As the difference is so small, I would say it is not a good problem of this sort-the answer is supposed to be "obvious" once you see it. – Ross Millikan Apr 22 '14 at 03:46
  • @RossMillikan But they are the same side!!! But I got what you wanted to mean. – Man_From_India Apr 22 '14 at 03:49
  • Your solution has them on the same side, which I see as better evidence for your answer. I was trying to show some things which feed into the evaluation of candidate answers. – Ross Millikan Apr 22 '14 at 03:50
  • The "10" does not come out of nowhere: the intended rule is that the product of the three numbers outside the triangle divided by the interior number is always 10. However, Man_from_India has found a consistent "rule" for the examples shown that justifies an answer of 30 . Obviously, this is an interpretation not anticipated by the poser of the problem, though it is less simple than what appears to be the intended reading. This isn't the first of these "inductive" puzzles that I've seen which are potentially ambiguous. Unfortunately, these silly things often show up on admissions tests. – colormegone Apr 22 '14 at 03:51
  • @RossMillikan Yes, I got it :) – Man_From_India Apr 22 '14 at 03:51
  • As we don't like unanswered questions here, please write up your answer and accept it. I think you have to wait a little while to accept it. You clearly understand the issue. I don't think there is a clear answer to the original problem. – Ross Millikan Apr 22 '14 at 03:59

1 Answers1

1

Thanks to Ross Millikan and DonAntonio for the discussion on this. And I got the answer. Here it is.

As for the answer I suggested -

This is not an universal answer. What I mean to say is that if in case of the 2nd triangle the middle number was changed to 35 for example, my suggested formula didn't work. In that case I had to use numbers from different sides of the triangle that shouldn't be identical for the case of other triangles.

But the solution of the book is universal, as it is.

I got the answer now.

Even saying so, the question itself is ambiguous.

Man_From_India
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