Question

Surjo is four years old and he is learning to write numbers. His math

notebook looks like a square grid with 20 rows and 20 columns. He usually writes the

numbers from top to bottom and when one column is finished he starts writing along the

next column. One day he starts writing the numbers from left to right (along the rows).

How many of the numbers will be placed in exactly the same place where it would have

appeared if he had written along the columns?​

Answer 1

Let n be such a number which remained in the position in both of the writing methods.

Let x and y be the row and column number of n, respectively, 1 ≤ x, y ≤ 20.

Then following the order of the numbers in the vertical writing method,

n = 20(y − 1) + x

Again by the horizontal writing method,

n = 20(x − 1) + y

⠀⠀⠀⠀$\sf{∴ 20(y − 1) + x = n = 20(x − 1) + y</p><p>}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀$\sf{ ⇒ x = y</p><p>}$

So, x must be equal to y and there are 20 such pairs. So they correspond to 20 possible values for n.