Unknown number of rows below staircase I have a staircase which looks like this: ![Staircase]( Chelsea adds a certain number of rows on the bottom of this staircase and then numbers the squares, like this: ![Filled staircase]( For the previous example we're assuming that Chelsea added 2 rows below the staircase. The top number in the 15th column is 405. How many rows did Chelsea add below my staircase. I worked out that the 15th column would have the same number of squares as the 15th odd number, which is 29, in a _Normal Staircase_. However, I don't know how to go about this. Can anyone help me?

Without adding rows, the first column has 1 stair, the first two columns, 4 stairs, the first three columns 9 stairs,... The first 15 columns have $15^2=225$ stairs. (Adding 2 each time starting at one gives odd numbers, the sum of the first n odd numbers is $n^2$.)

$405-225=180$ stairs are in the **extra** rows spreading beneath the 15 columns, so $180/15=12$ rows.