A Mathematical Investigation

How to Investigate Dr Amber Habib Mathematical Sciences Foundation St. Stephen’s College Delhi 110007

I. Understanding the Problem We have to identify the parts:

1. What is the unknown?

2. What are the data?

3. What conditions must be met?

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Also:

1. Create suitable notation.

2. Draw a figure.

3. Separate the various parts of the condition.

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An Example Take a natural number, such as 24933. Add its digits, so in this case you get 2 + 4 + 9 + 3 + 3 = 21. Repeat the process with the new number. Repeat till you get a single digit number. Our example stops when we get 3. Question: What happens if we start with a square? Examples: 1 → 1, 4 → 4, 9 → 9, 16 → 5, 25 → 7, 36 → 9, 49 → 13 → 4, 64 → 10 → 1, 81 → 9, 100 → 1

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A Conjecture If we apply the above process to a square, we get either 1,4,5,7 or 9. Now what can we do with this conjecture? Let’s create suitable notation. We will use letters such as m, n for natural numbers. If we start with n and carry out the process described earlier, then the result will be called digit(n). Thus, digit(25) = 7, digit(49) = 4, etc. And the conjecture is: For any natural number n, digit(n2) = 1, 4, 5, 7, or 9 4

II. Devising a Plan We have to connect the data with the unknown.

1. Do we know a related problem? Could we use its result or its method?

2. Specialize: Can we imagine a more special problem?

3. Generalize: Can we imagine a more general problem?

4. Can we solve a part of the problem? 5

5. Go back to definitions.

6. Work backwards: Can we imagine what data would suffice to solve the problem? Can we transform the problem so the data and the unknown are more clearly related?

7. Have we tried the standard techniques: exhaustion, contradiction, induction, counting.

8. Have we used all the data? Considered all the conditions?

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Creating a Plan for our Conjecture We should decide whether our initial line of attack will be to prove or disprove the conjecture.

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A Fact The conjecture was tested by a computer program for the square of every number up to 10,000. So it is reasonable to try to prove it. Where should we start?

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One can start from various directions:

1. Squaring is a special case of multiplication, so we could study digit(mn) (We are “generalizing”!). Note that multiplication, in turn, is a special case of addition, so we could study digit(m + n) (We are also “returning to definitions”).

2. Since we have to prove something for all natural numbers, we could consider trying Mathematical Induction. Then we have to deal with the expression digit((n + 1)2) = digit(n2 + 2n + 1) which brings us to the same conclusions as before. 9

The outlines of a plan are now taking form. Let’s run a couple of examples (“specialize”): digit(120 + 329) = digit(449) = digit(17) = 8 digit(120) = 3 digit(329) = digit(14) = 5 digit(123 + 329) = digit(452) = digit(11) = 2 digit(123) = 6 digit(329) = digit(14) = 5

Notice anything?

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The first example suggested the possibility that digit(m + n) = digit(m) + digit(n), but the second example makes us adjust that to digit(m + n) = digit(digit(m) + digit(n)).

Before we ask whether this is true or not, let’s find out: Is it even useful? What will it say about products, and hence about squares?

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Assume that digit(m + n) = digit(digit(m) + digit(n)) is always true. Then: digit(mn) = digit(n · · + n}) | + ·{z m times = digit(digit(n) + ·{z · · + digit(n)}) | m times = digit(m digit(n)) = digit(digit(m) digit(n))

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Thus, assuming that digit(m + n) = digit(digit(m) + digit(n)), we also find digit(mn) = digit(digit(m) digit(n)), and in particular: digit(n2) = digit(digit(n)2). Now digit(n) is a single digit number, so to find the possibilities for digit(n2) we merely have to run through the cases n = 1, 2, . . . , 9. That we have already done! (Thus our plan started by generalizing and ended by specializing.)

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III. Carrying Out The Plan

1. Check each step.

Let’s go over our example once more... Did you find any steps that are not clear? That you think need proof?

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Looking Back and Forward Examine the solution.

1. Can we check the result? Can we check the argument?

2. Can we derive the result differently?

3. Can we use the result, or method, for other problems?

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References

1. G. Polya, How to Solve It. Second Edition, Prentice Hall India.

2. Steven G. Krantz, Techniques of Problem Solving, Universities Press (India).

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