Math Explorations

These are open-ended explorations of math relations without any one correct answer. We follow certain procedures which may end up in a surprising result. Many such explorations, which are part of Number Theory, have been done by mathematicians and it would be interesting for students to try them out. We give a few examples.

Making numbers

1. Select any three one-digit numbers, preferably all different.

2. Form as many 1, 2- or 3-digit numbers with these 3 numbers without using any number only once.

3. Arrange them in ascending order.

   1. This gives practice in forming numbers and comparing their values.

   2. The student may also see that the total number of possible combinations is independent of the initial number (as in quantity) of one-digit numbers.

4. This exercise also introduces students to elementary ideas in combinations.

Odds & Evens

1. Write down a number (in initial stages write a small number to get practice)

2. Work out the next number as per the following rules

   1. If the number is even, then divide it by 2

   2. If the number is odd then multiply it by 3 and add one. In short find 3n+1

3. Write down the next number below the previous number

4. Repeat steps 2 & 3 until you come to a surprising end.

This procedure & its result are famous in math & are described in the chapter 32.6 on "Famous Conjectures".

Kaprekar's Constant

1. Write down any 4-digit number

2. Write the biggest number possible with the 4 digits

3. Write the smallest number possible with the 4 digits

4. Find their difference (you should arrive at another 4-digit number)

5. Repeat steps 2 to 4 until you come across a surprise ending.

For any 4-digit numbers, it usually takes about 7 steps to arrive at the constant.

This number behaviour was discovered by an amateur Indian school-teacher-mathematician called D R Kaprekar.

Kaprekar had an experience very similar to that of Ramanujan several decades before him. Mathematicians in India did not take his work in number explorations seriously for a long time.

In March 1975, Martin Gardner wrote about Kaprekar in his March 1975 column of Mathematical Games for Scientific American. After this his work became widely known.

Does it work with a 3-digit number?

Try it out!

Describing Numbers

1. Write a 3-digit number

2. Count the number of times, the various numerals appear in that number

   1. If the number chosen was 348, the numeral 3 occurs 1 time, 4 1 time and 8 1 time.

3. Write the description of the above number in this form

   1. 131418 (the numerals 3, 4 & 8 must be written in the ascending order and the number (quantity in this case 1, should be written before the numeral)

4. Repeat Step 3

   1. Here the numerals are 1, 3, 4 & 8

   2. The number of times each appear are 3 in case of 1, 1 in case of 3, 4 & 8.

   3. Hence the describing number is 31131418(the numerals, shown in bold and the number of times they appear are written before the numeral.

5. Repeat Step 3 until you get to a surprising end.

Though this may look complicated in text, it is very easy when understood. This exercise gives practice of understanding and following an algorithm.

Many more examples can be found in <"http://www.openmiddle.com"> for all grade levels.