S K Gupta and Anubhuti Gangal Solutions for Chapter: Playing with Numbers, Exercise 3: Exercise 5(C)

Write the nine non-repetitive digits in the circles of this triangle in such a way as to have a total of $20$ on each side. 

The magic hexagon has a total of $19$ hexagons within it. All sets of $3, 4$ and $5$ hexagons in a row add to $38$. Fill in the missing figures. No number should be repeated. 

Rumours: Rumours spread very fast. Sometimes an incident witnessed by just a few persons becomes the talk of the town within less than two hours. Suppose a man comes at about $8 \mathrm{a}.\mathrm{m}.$ to a town with a population of about $50000$ and conveys an interesting news to just three persons. This takes up, say, $15 \mathrm{minutes}$. Within $2 \mathrm{hours}$ it becomes a talk of the town. Show, how is it possible. 
 

Number patterns:

$0\times 9+1=1$

$1\times 9+2=11$

$12\times 9+3=111$

$123\times 9+4=1111$

$1234\times 9+5=11111$

$12345\times 9+6=111111$

$123456\times 9+7=1111111$

$1234567\times 9+8=11111111$

$12345678\times 9+9=111111111$

$123456789\times 9+10=1111111111$

Investigate a similar number pattern where the first two lines are:

$1\times 8+1=9$

$12\times 8+3=98$

Choose any four digits, for example $6, 8, 9$ and $2$.

Repeat the first digit at the end.

Find the difference between the pairs of digits next to each other. Always repeat the first digit at the end.

Continue the pattern untill all the numbers are same.

Try the above pattern these numbers: $a. 4, 5, 5,$ $2$ $b. 9, 4, 3, 8$ $c. 4, 8, 7, 6$ and $d. 8, 6, 3, 2$

• Enter any three digits in your calculator.                    $753$

• Repeat the three digits to get a six-digit number.       $753753$

• Divide by $13$.                                                             $57981$

• Divide by $11$.                                                             $5271$

• Divide by $7$.                                                                 $753$

• You will find that the result is your original number.

Try the pattern with the numbers $347, 952, 885$ and $409$.

Blaise Pascal was a famous French mathematician and philosopher. One of his most important discoveries is known as Pascal's Triangle. See if you can discover the secret of this pattern of numbers. 

Solve:

Write row $8$ of Pascal's Triangle.

Write row $9$ of Pascal's Triangle.

Write row $10$ of Pascal's Triangle.

What is the sum of the numbers in row $1$ of  Pascal's Triangle.

What is the sum of the numbers in row $3$ of  Pascal's Triangle.

What is the sum of the numbers in row $8$ of  Pascal's Triangle with adding.

Sudoku: Fill in the grid so that every horizontal row, every vertical column and every $3\times 3$ box contains the digits $1‐9$, without repeating the numbers in the same row, column or box. You can't change the digits already given in the grid.