Given that a and b are non-negative integers and f(x,y)=(x+y-a)(x+y-b), then if f(x,y)=0 has n distinct non-negative integer solutions for (x,y), find how many different polynomials f(x,y) can take.
Note: For example, (1,0) and (0,1) are not distinct solutions.
Mary has an unusual ring. It is made up of a set of three interlocking ringlets, each set with a large gemstone, a diamond, a ruby, and a sapphire. It is very valuable and the insurance company insists it be kept locked in their vault, except on those occassions where Mary is wearing it. Because of this, Mary had a copy made, which she wears on lesser occassions.
Today she attended an affair in which she was able to convince the insurance company that she needed to wear the original. Half an hour after returning home the insurance rep called to let her know he was on his way to pick up the ring.
Mary then realized that she'd absent-mindedly taken off the ring and put it in her jewelry box, where she keeps the copy.
When it's not on her finger, the ring separates into its componants, so Mary was looking at six nearly identical ringlets, two with blue stones, two with red, and two with white.
She needs to separate the genuine ringlets from the copies. She knows that each of the ringlets in each set weighs the same.(That is the genuine ringlets each weigh the same, and the copies each weigh the same.) She also knows that a copy weighs less than its original.
The only scale she has that is delicate enough to properly weigh the ringlets is a small balance scale she uses to measure headache powders and sleeping draughts (she can't swallow pills), but the weights tha she uses for her medicine are of an order too small for the ringlets. She will have to weigh them against each other. She could do it in three weighings by trying each against its counterpart, but she is certain to be "caught" before she finishes, and either her insurance will go up, or the company will be "forced" to not allow her to wear the ring any more.
Is there a way to separate the six ringlets in two weighings?
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