When I see a good problem somewhere, I like to investigate its properties further and deeper, with the intention of using it to develope problem solving skills in my students. Such was the case when I saw the two problems that you will see in this page of WTM. I hope you will agree with me that they can be useful to bring inter-esting math challenges to young students at the upper elementary levels (3rd-6th).
The first problem appeared in the 1994 MATHCOUNTS contest exams (School level, Sprint #5). It said:
Two positive numbers are such that their difference is 6 and the difference of their squares is 48. What is their sum?
The foundation concept of this problem is a perennial topic in all high school Algebra I courses: the difference of two squares pattern. It occurs in the chapters on multiplying and factoring binomials. Solving such a problem in an algebra course is, there-fore, a somewhat regular, if not trivial, matter.
A possible solution process might go as follows:
1. x2 – y2 = (x + y)(x – y)
2. x – y = 6 and x2 – y2 = 48
3. 48 = (x + y)(6)
4. x + y = 8
However, elementary students are not expected to work at such an abstract level of thinking. But if they have access to calculators and a little basic guidance in understanding what the problem is all about, they can enjoy a meaningful experience just the same. We proceed by setting up a t-chart to organize our work.
Filling out the entries — by educated trial and check — now becomes an easy task. In fact, for this MATHCOUNTS problem it is a rather quick one: A = 7 and B = 1; thus the sum is 8. This is merely because it was part of a large set of problems to be solved under a time limit. Hence it was
not intended to be a hard, time-consuming item. Also it should be pointed out that calculators are not allowed on this portion of the contest.
However, if number size is increased (moderately at first) and time is removed as a factor, many exercises can now be formulated. Here are some examples:
1. Two positive numbers are such that their difference is 6 and the difference of their squares is 180. What is their sum?
2. Two positive numbers are such that their difference is 7 and the difference of their squares is 161. What is their sum?
3. Two positive numbers are such that their difference is 10 and the difference of their squares is 260. What is their sum?
4. Two positive numbers are such that their difference is 15 and the difference of their squares is 555. What is their sum?
1. A = 18, B = 12, & sum = 30.
2. A = 15, B = 8, & sum = 23.
3. A = 18, B = 8, & sum = 26.
4. A = 26, B = 11, & sum = 37.
The work on these problems can be made a lot easier and more efficient if one uses certain special features of calculators. First, if one is using an ordinary 4-function nonscientific model, here is an interesting shortcut method that takes advantage of the memory keys. Using the answer of #4 above, the method goes this way:
0. Make sure the memory register is clear.
1. Press: 26, [x], [M+]. (This puts A2 into the memory.)
2. Press: 11, [x], [M-]. (This computes the square of B and subtracts it from the value from A2.)
3. Press: [MR] (or [MRC]). (This shows the difference.)
Of course, if one is using a regular scientific model, the steps are even shorter and memory need not be utilized to obtain the same results. The key sequence would be as follows:
26 [x2] [-] 11 [x2] [=]
This problem likewise appeared as part of the same MATHCOUNTS contest; it was #23 on the Sprint round.
What is the smallest multiple of 5 the sum of whose digits is 18?
We should remind ourselves once again that this is a timed contest and no calculators permitted. Hence, one might be expected to solve this question analytically, perhaps as follows:
“Since all multiples of 5 end in 5 or 0, and our desired multiple must contain at least 3 digits, we are looking for a value in one of these two forms:
aa0 or bc5. The only number in the first form to have a digital sum of 18 is 990. But it could not be the smallest one because the b-c digits of the
second form will certainly be smaller due to the help of the 5. Of course, the sum of the b-c digits will then be 13, the only possibilities being 4 & 9, 5 & 8, and 6 & 7. Therefore, the smallest multiple will be produced by the pair containing the smallest digit, which is 4 & 9. So the problem’s answer is 495.”
It might be noted here that a new question can be asked using the facts presented in the above solution. It would be:
How many multiples of 5, less than 1000, have a sum or their digits that is 18?The answer of seven is easily seen by making a list like this:
495 585 675 990 945 855 765
But now let’s bring this whole situation down to a basic, more elementary level, one that uses calculators, mental addition, and emphasizes more strongly the concept at the heart of the original problem, multiples of a number. We might proceed as follows:
1. Present the problem with as little initial explanation as possible and allow the student time to wrestle with it.
2. Then as necessary, direct the student to use the calculator to produce a series of multiples of 5 by utilizing its constant addition feature. [For many nonscientific models, this simply means pressing “5”, [+], [=], then continuing with the [=] key as often as needed.
3. It is here that the attack could take two different directions: (a) making a t-chart of the multiples and their digital sums, observing patterns along the way; or (b) simply adding the digits mentally, and as rapidly as possible, as one presses [=]. Each strategy has its positive and negative side; the learner should choose whichever way seems best.
It should be obvious that many more problems of this nature can be posed by changing either the multiples’ factor, the digit sum, or both. Here is an example of each style:
1. What is the smallest multiple of 6 the sum of whose digits is 18? [Ans. 198]
2. What is the smallest multiple of 5 the sum of whose digits is 15? [Ans. 195]
3. What is the smallest multiple of 7 the sum of whose digits is 16? [Ans. 196]
This is a prime example of how one simple problem can be turned into many, and in which important concepts are present and yet basic skills can be practiced. Additionally, it is a case where students could be encouraged to invent their own problems, thus becoming a more integral part of the learning process, a factor often overlooked in many math classrooms today.