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 ( ) / \   / \ ( ) --  -- ( ) Q2 ( ) / \  [-12] / \ ( ) --  -- ( ) 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 http://mathforum.org/library/drmath/view/57520.html 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 - x) --  -- (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) / \   / \ (20.5) --  -- (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) 73! 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) / \ [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 http://mathforum.org/dr.math/
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.
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