[Math] Determine the value of N

algebra-precalculus

I'm a complete n00b at math, but I'm wondering how one would go about determining the value of n in the following comparison.

n * 1.5 + 12.5 = 12.5 / 2 + n

I'm new to the math StackExchange, so I'm also not sure how to properly format this question. Feel free to edit.

$1.5n+12.5=\displaystyle \frac{12.5}{2}+n$

Explanation

I don't think I formulated my mathematical equation properly because I know that the value I'm looking for is obviously a positive integer.

I'm trying to figure out what size the squares must be, so that the center most square in each row is centered above or below the gap between the two squares in the other row.

  1. All the squares must be the same size.
  2. All of the gaps are 12.5 pixels.

The squares in the image below are obviously not big enough as of right now.

enter image description here

Best Answer

Let the length of a gap be $g$ and the distance between the white lines be $l$. The distance of the left edge of the left square of the top row (in the image) from the white line on the left is$$n+\frac g2-\frac n2=\frac{n+g}2$$Adding to this the length of the $k$ squares in the top row and the $(k-1)$ gaps between them,$$\frac{n+g}2+kn+(k-1)g=l\\\implies\left(k+\frac12\right)n+\left(k-\frac12\right)g=l\\\therefore n=\frac{l-\left(k-\frac12\right)g}{k+\frac12}$$For finding $n$, just substitute the value of $k$.

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