Equals What?
(original posting of this problem was incorrect ... here is the corrected version)
492x04 = [3(230+y)]^2
Solve for x and y.
492x04 = [3(230+y)]^2
Solve for x and y.
posted by Zaux at
5:34 AM
Sunday, January 10, 2010
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11 Comments:
x=8, y=4
Assuming that x and y are integers, is insufficient to provide a unique answer. So anticipate that x and y are going to be positive integers.
The RHS is exactly divisible by 9 (in the integer domain, obviously), so the LHS must be divisible by 9 also. In base 10 notation, if an integer is divisible by 9, then the sum of the digits is also divisible by 9. So 4+9+2+x+0+4 = 19+x must be divisible by 9. Only x = 8 will do that.
So, 492804 = 9*(230+y)^2
=> 54756 = (230+y)^2
Taking square roots and swapping LHS with RHS => 230 + y = +/- 234
=> y = -230 +/- 234
=> y = -464 or 4
I expect that the desired solution is x = 8, y = 4.
Hi Miguel, long time no see.
LOL. I was posting a long diatribe about how to do this, when the original problem got pulled. About the only thing one can assume (without using the unwritten laws and conventions with puzzles) is that x is an integer, otherwise 492x04 would be bizarre by most standards. But that would give 10 possible values for x, and for each of those, 2 values for y. As this is a puzzle website, one invokes the rule that y is a positive integer, because that assumption gives the nicest way of defining a unique solution (which is what I take for granted is being sought). I've been unusually explicit in stating the assumptions that I've used.
If allowed that x wasn't an integer, then the elegant way of showing that x = 8 would be lost, and the problem would, in my opinion, have little intellectual value.
.. oops, I really meant to have put more emphasis on the importance of assuming y to be an integer in my last post.
Hi Chris. I confess I've been overworking... *sigh*
Guys, once again, I apologize for the error in the original post. I was pretty tired and it was in the wee hours of the morning. I don't normally transpose numbers ... heh heh!
Hi Cam. I agree, the previous blog should have been left as it was a good chance to demonstrate puzzle problem solving thinking more explicitly.
NB. That's 20 values of y not 18. You need more coffee ;)
Hi Zaux. Don't worry about it. As I don't think I've said it for a while, my favourite (mis)quote is: "the man who made no mistakes, made nothing".
I'll second that. Having been a semi-frenzied problem poster a few months back (but can't hold a candle to Zaux), I know it takes quite a lot of dedication and time. I also know that it's easy to speed-read a problem and solution and post before checking that the problem and solution was good. Surprisingly often, mistakes aren't always a bad thing - they can really test the mettle of the would-be solver.
Thank you for all your hard work Zaux.
this is actually quite simple if you solve it for "y".
y = ((sqrt(492x04))/3)-230
afterwards, plug in numbers for "y" starting from 0 and you get it to equal 4 and x to equal 8
Hi Puja. You've solved for y, that was a good move as there are only 10 possible values for x. But your instructions were the wrong way round:
Afterwards, plug in numbers for x starting from 0 and you get it to equal 8 and y to equal 4 - makes much more sense.
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