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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?

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    $\begingroup$ I lack information what a staircase or even a Normal Staircase is. $\endgroup$ – mvw May 10 '17 at 11:51
  • $\begingroup$ Without adding rows, there will be $225=15^2$ stairs up to the top of the 15th column. Now $15x+225=405$, no? So 12 rows? $\endgroup$ – Arby May 10 '17 at 11:55
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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.

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Let's say that Chelsea added $n$ rows below your staircase. Prior to this, the $k$th column of your staircase had $2k-1$ stairs in it, and now has $2k-1+n.$ So, the first $15$ columns of your staircase have a total of $$405=(1+n)+(3+n)+\cdots+(29+n)=15n+1+3+\cdots+29$$ steps in them. You need only solve for $n.$

See if you can find a pattern to the following numbers: $1,1+3,1+3+5,1+3+5+7,....$ This should help you figure out what $1+3+5+7+\cdots+29$ is, and from there, solving your problem will be easy.

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Yes, the sum of the first $n$ odd numbers equals $n^2$, which is something you can prove with induction or a variety of other ways.

Thus, without any extra rows below the staircase, you would have $15^2=225$ squares. The top square of the 15th column has number $405$, and given its position in the diagram that means that the total number of squares in the 15 columns is $405$ as well. So that means that there are $405-225=180$ extra squares, divided evenly over the extra rows below the staircase. So: with $x$ extra rows, you get $15x=180$, so $x=12$

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