Finding the OperationDate: 06/21/99 at 22:53:08 From: don yehling Subject: Series problem This is from Max Sobel's _Teaching Mathematics_ (page 28.) (I have substituted ? for * to avoid confusion with multiplication.) The object is to find the operation '?'. Given: 3 ? 4 ... 5 4 ? 7 ... 1 8 ? 4 ... 0 1 ? 2 ... 9 Then he asks: 5 ? 5 ... 4 ? 1 ... 6 ? 2 ... I find that the first three are 32 minus the product of the two numbers divided by four, but my solution breaks down with the fourth pair of numbers. Sobel adds that this problem came from a third grade math text, so you have to use operations a third grade student would be familiar with. I'm stumped. Date: 06/22/99 at 02:34:30 From: Doctor Pete Subject: Re: Series problem Hi, This is an interesting problem. No doubt you noticed that the largest pair, 8?4, gives the least result, 0, and the smallest pair, 1?2, gives the greatest result, 9. Clearly we are dealing with a function where subtraction is involved. The next hint you provided was that this is a "third grade" level problem, so the key here is not to look too hard. So let us not consider division. A bit of computation shows 3 + 4 = 7 3 * 4 = 12 4 + 7 = 11 4 * 7 = 28 8 + 4 = 12 8 * 4 = 32 1 + 2 = 3 1 * 2 = 2 Clearly these are good places to start because the operations of addition and multiplication preserve the property that the larger the given pair, the larger the result. But we note that in the latter case, were we to subtract these values from a given number, the differences between the results would be too large. On the other hand, a bit of thought shows that 12 - 7 = 5 12 - 11 = 1 12 - 12 = 0 12 - 3 = 9 And so we have found the rule: Take the two given numbers, add them, and subtract the sum from 12. Algebraically speaking, the binary operation x ? y is given by the function x ? y = f(x,y) = 12 - (x+y). It is now straightforward to find 5?5 = 12-(5+5) = 2, 4?1 = 12-(4+1) = 7, 6?2 = 12-(6+2) = 4. Now, I say this problem is interesting, because it demonstrates an important method of thinking: that of understanding how certain familiar operations (addition, subtraction, etc.) can be thought of as actions on "inputs," and furthermore, how the result depends on the sizes of the inputs for a given operation. It also introduces on an intuitive level algebraic concepts, where there is some sort of "reverse" thinking involved. - Doctor Pete, The Math Forum http://mathforum.org/dr.math/ |
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