Using symmetry
by rrusczyk, Jun 17, 2006, 3:21 PM
A problem from this week's San Diego Math League for our younger students read something like
Given
$\begin{eqnarray*} 2x + 5y &=& 48\\ 5x + 2y &=& -13 \end{eqnarray*}$
Find
.
I really like this problem for younger students for a couple reasons. First, it's accessible to any student who has learned basic systems of linear equations. Second, there's a very slick and quick solution that illustrates a couple very effective general problem solving principles.
More experienced readers will see the solution quickly. Add the given equations to get 7x + 7y = 35, so we know that x + y = 5. We can now find x and y quickly, but why bother? We want 2005x + 2005y = 2005(x+y). We have x+y, so our answer is 2005(5) = 10025 and we're done.
Some may call this a cheap simple trick, but it's an example of a very powerful approach. When a problem has symmetry - use it!
In this one, the left hand sides of our given equation have symmetry, so we might think to add them to get x + y, in which our variables are clearly symmetric. Our big hint to go after symmetry is what we're looking for: 2005x + 2005y. Clearly this is a nice, symmetric expression.
This brings us to our second problem solving tip: keep your target in mind! When you first see this problem, you might think to go ahead and solve for x and y. That works fine, but when you take a look at what you seek, you see a nice symmetric expression. Brain goes 'ding' and you see that you can use the given equations to get what you want.
Here are a couple other symmetry problems.
1) Find
if
$\begin{eqnarray*} v+w &=& 6\\ w+x &=& 7\\ x+y &=& 13\\ y+z &=& 20\\ z+v&=& -3 \end{eqnarray*}$
2) The areas of 3 faces of a rectangular box are 45, 56, and 21. Find the volume of the box. Can you find the dimensions?
Please add some symmetry problems of your own!
Given
$\begin{eqnarray*} 2x + 5y &=& 48\\ 5x + 2y &=& -13 \end{eqnarray*}$
Find
.I really like this problem for younger students for a couple reasons. First, it's accessible to any student who has learned basic systems of linear equations. Second, there's a very slick and quick solution that illustrates a couple very effective general problem solving principles.
More experienced readers will see the solution quickly. Add the given equations to get 7x + 7y = 35, so we know that x + y = 5. We can now find x and y quickly, but why bother? We want 2005x + 2005y = 2005(x+y). We have x+y, so our answer is 2005(5) = 10025 and we're done.
Some may call this a cheap simple trick, but it's an example of a very powerful approach. When a problem has symmetry - use it!
In this one, the left hand sides of our given equation have symmetry, so we might think to add them to get x + y, in which our variables are clearly symmetric. Our big hint to go after symmetry is what we're looking for: 2005x + 2005y. Clearly this is a nice, symmetric expression.
This brings us to our second problem solving tip: keep your target in mind! When you first see this problem, you might think to go ahead and solve for x and y. That works fine, but when you take a look at what you seek, you see a nice symmetric expression. Brain goes 'ding' and you see that you can use the given equations to get what you want.
Here are a couple other symmetry problems.
1) Find
if$\begin{eqnarray*} v+w &=& 6\\ w+x &=& 7\\ x+y &=& 13\\ y+z &=& 20\\ z+v&=& -3 \end{eqnarray*}$
2) The areas of 3 faces of a rectangular box are 45, 56, and 21. Find the volume of the box. Can you find the dimensions?
Please add some symmetry problems of your own!
and 
.
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,
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February 2012
January 2012