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11. Here is a magical hexagon whose magical constant is 38. Colour the supercells such that the numbers
in any adjacent cells (horizontal, slanting) are less than the number in supercell.

Digit Sums of Numbers

Nupur observes that when she adds up the digits of certain numbers, the sum is the same. For
example, adding the digits of the number 75 will be same as adding the digits of 246 or 3018.
7 + 5 = 12; 2 + 4 + 6 = 12; 3 + 0 + 1 + 8 = 12

In how many ways can you write 12 as a sum of digits using from 0 to 9?
Let’s try: 12 = 3 + 9 = 4 + 8 = 5 + 7 = 6 + 6 = 1 + 3 + 8 = 2 + 3 + 7 = 3 + 4 + 5 = 4 + 4 + 4 = 5 + 2 + 5 ...
many combinations are possible.
Thus, for a given digit sum, many numbers of different digits can be written. Here, for the digit
sum 12, the smallest two-digit number is 39 and the largest two-digit number is 93; the smallest
three-digit number is 129 and the largest three-digit number is 930.

Quick Check
Maths Talk
1. Write a few 4-digit numbers whose 1. How big a number can you form with
digits add up to 12. the digit sum 12? Will the greatest number be
2. What is the smallest 5-digit number whose even or odd?
digits sum is 12? 2. For the digit sum of 12, write all the three-digit
3. What is the largest 6-digit number whose digits numbers with digits in ascending order.
sum is 12? Share your observations with the class and discuss.

Enrichment
Do you know the importance of digit sum in mathematics? Let us explore the rule of nine or casting out nine
method. In Vedic Mathematics book it is called – Navansh (uoka'k). The basic meaning of Casting out Nines is to
throw out Nines. This method is applicable in checking the fundamental operations in mathematics. Let’s first
check how it works:
Digit sum of 2314 = 2 + 3 + 1 + 4 = 10; Digit sum of 10 = 1 + 0 = 1
Since 2 + 3 + 4 = 9 and if you apply casting out nine method, then the pair/group whose sum is 9 should be
removed.
Hence, the digit sum of 2314 is (2 + 3 + 4 ) + 1 = 1, as we removed 2 + 3 + 4 = 9
Let us take another number: 1729
Its digit sum is 1 + 7 + 2 + 9 = 19 and 19 being a two-digit number, will give 1 + 9 = 10 that will finally result as
1 + 0 = 1. Hence, the digit sum of a number should lie from 0 to 8.
Take 10 three-digit numbers of your choice, no digit is repeated.
Now, calculate its digit sum and write it in a tabular form. Colour the supercell.

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