Math Explorations

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These are open-ended 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

Select any three one-digit numbers, preferably all different. Form as many 1, 2- or 3-digit numbers with these 3 numbers without using any number only once. Arrange them in ascending order. This gives practice in forming numbers, comparing their values. The student may also see that the total number of possible combinations is dependent of the initial number (as in quantity) of one-digit numbers. This exercise also introduces students to elementary ideas in combinations.

Odds &amp; Evens

This procedure &amp; its result are famous in math &amp; are described in the chapter 32.6 on &ldquo;Famous Conjectures&rdquo;.
 * 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&rsquo;s Constant

This 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.

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 <a href="http://www.openmiddle.com"> www.openmiddle.com </a>for all grade levels.

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