Akarsh Gupta (Class VII, A.P.J., Noida), Sarthak Parikh (Class VIII, Sardar Patel) and Richa Sharma (Class IX, Swami Vivekananda Sarvodaya School) all gave excellent explanations/solutions to the questions asked in the previous column. The answers (very briefly) are:
Q1. The numbers that have $0, 2, 4, 6, 8$ in the last digit are divisible by $2$.
Q2. The numbers that have $0$ or $5$ in the last digit are divisible by $2$.
Q3. If the sum of the digits of a number is $9$ or a multiple of $9$, then that number is divisible by $9$.
Q4. If the sum of the digits of a number is a multiple of $3$ then that number is divisible by $3$.
Sarthak Parikh also asked an interesting question. Sarthak asked: How many lines of symmetry does a parallelogram have? What about the square? Rhombus? Rectangle? Octagon? Cut these shapes out of paper and label the vertices. Experiment by drawing various lines, and flipping the shape. If you get back the same shape (but with different labels) you have discovered a line of symmetry. For example, a square with vertices labeled ABCD, you may get a square labeled BACD (see the top two squares in the picture). If you notice, by flipping across a line you have rotated the square by 180 degree. You should also try to find other rotational symmetries. When you rotate a square ABCD clockwise by 90 degrees, you get the square DABC (bottom square in the picture). After doing some experiments, perhaps you would like to answer Sarthak’s query?
Here is another activity involving squares—the squares of natural numbers.
Complete the following table. For n going from 1 to 20 write the square of n in the second column, and the sum of the first n odd numbers in the third column. The odd numbers are (1, 3, 5, …).
$n$

$n^2$

Sum
of first $n$ odd numbers

$1$

$1$

$1=1$

$2$

$4$

$1+3=
4$

$3$

$9$

$1+3+5=9$

…



$20$



Make a picture showing the pattern above. (Hint: see the picture below)
 Find a formula for the sum $O$ of the first $n$ odd numbers: $O = 1+3+5+7+\cdots +(2n1).$
 Use the formula for $O$ to find a formula for the sum $E$ of the first $n$ even numbers: $E=2+4+\cdots+2n .$
 Use the formula for $E$ to find a formula for the sum of the first $n$ natural numbers.
Check your work by putting $n=1, 2, 3, 4, 5$ for each formula you find.