Math Explorations

< 31.2 Folk Mathematics | Topic Index | 31.4 Magic Squares >

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.
 * 4) This gives practice in forming numbers and comparing their values.
 * 5) 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.
 * 6) This exercise also introduces students to elementary ideas in combinations.

Odds & Evens

This procedure &amp; its result are famous in math &amp; are described in the chapter 32.6 on "Famous Conjectures".
 * 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 				If the number is even, then divide it by 2If 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 &amp; 3 until you come to a surprising end.

Kaprekar's Constant

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

Does it work with a 3-digit number?

Try it out!

Describing Numbers

Though this may look complicated in text, it is very easy when understood. This exercise gives practice of understanding and following an algorithm.
 * 1) Write a 3-digit number
 * 2) Count the number of times, the various numerals appear in that number 				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 				131418 (the numerals 3, 4 &amp; 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 				Here the numerals are 1, 3, 4 &amp; 8</li>The number of times each appear are 3 in case of 1, 1 in case of 3, 4 &amp; 8.</li>Hence the describing number is 31131418(the numerals, shown in bold and the number of times they appear are written before the numeral.</li></ol>
 * 5) Repeat Step 3 until you get to a surprising end.

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

< 31.2 Folk Mathematics | Topic Index | 31.4 Magic Squares >