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Magic Triangle Puzzles, Expanded

Date: 11/17/2012 at 16:15:30
From: Sanjeev
Subject: Magic Triangle Puzzle

My daughter is in year 6 of the UK school curriculum, and we're trying to
solve these two magic triangle puzzles:

Q1                (  )
                /      \
            [34]        [39]
            /              \
         (  ) --  [46]  -- (   )

Q2                (  )
                /      \
            [13]       [-12]
            /              \
         (  ) --  [15]  -- (   )

I have looked up the solution to a similar maths puzzle in your archives.
However, the sum of the three numbers in the first puzzle is 119, which,
when divided by 2, gives the decimal number 59.5. How do I continue to
work on the puzzle? And the second question has a negative numeral. How
does one work that one?

Date: 11/17/2012 at 17:58:05
From: Doctor Peterson
Subject: Re: Magic Triangle Puzzle

Hi, Sanjeev.

You didn't say what the rules are for this. From the title "magic
triangle," I could infer that the sum of numbers on each side has to be
the same, as in magic squares. But probably it is the kind of puzzle where
the numbers on the sides have to be the sum of the numbers on either side
of them; and the following page is probably what you are referring to:

  Magic Triangle Puzzle  

Unless there is a typo, you are right that Q1 must involve non-whole
numbers. Clearly, Q2 involves negative numbers, which may or may not be
integers. So both are beyond the basic level, and are harder to do without
algebra, because there are more numbers to try.

Let's do them with algebra, then. I'll use a method that requires only one
variable, unlike Dr. Rick's method. I call one missing number x, and use
the rule governing the puzzle to fill in the other corners:

Q1                (x)
                /     \
            [34]       [39]
           /               \
     (34 - x) --  [46]  -- (39 - x)

Now the bottom row tells me that

   (34 - x) + (39 - x) = 46

Solving it, I find

   73 - 2x = 46
       -2x = -27
         x = 27/2
           = 13.5

The solution, then, is

Q1               (13.5)
                /      \
             [34]       [39]
            /               \
       (20.5) --  [46]  --  (25.5)

You can do the same for Q2.

I've seen several other tricks for solving these without algebra, but at
the cost of a lot of thinking to discover the trick. One of them may have
been suggested in your daughter's class.

Alternately, her teacher may expect an intelligent guess-and-check
procedure, one that involves picking a value for one corner, filling in
the others, and seeing what change to make to improve the result before
trying again.

For example, you could start by trying 0 at the top, and fill in the
bottom to make the sides add up:

Q1               (0)
               /     \
            [34]     [39]
           /             \
        (34) --  [46]  -- (39)

We need to reduce the bottom middle number by 27; what change in the top
number might do that?

Ordinary smart thinking will at least get you moving in the right
direction; really smart thinking gets you straight to the answer.

One trick I've seen can be discovered by doing a number of these puzzles,
usually with small numbers, and noticing a pattern (then confirmed by
looking for a reason it might always happen). This gives a quick way to
solve the puzzle. It was hinted at by coloring each vertex differently,
and coloring each side number to show the two colors being added there. We
can represent this with letters to show what is being added (without
having to think of them as actual variables):

                /   \
         [A + B]     [A + C]
            /            \
        (B) -- [B + C]  -- (C)

What do I get if I add two of the (given) numbers in brackets? What could
I do to get one of the unknowns out of that?

- Doctor Peterson, The Math Forum 

Date: 11/18/2012 at 19:21:01
From: Sanjeev
Subject: Thank you (Magic Triangle Puzzle)

Thank you so much.

I had a feeling that the solution would need decimals but wasn't sure....

Thank you.
Associated Topics:
Middle School Algebra
Middle School Puzzles

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