I recently thought of a math problem (or a series of math problems) that I can't solve. I would like to know of possible solutions or if it is impossible.
A square of arithmetic is a 2 by 2 grid:
$\array{a & b\cr d & c}$
Each cell is filled in with a rational number (or represented by a letter.)
For it to be valid, the following conditions must be met:
- $a + b = c$
- $b - c = d$
- $c \times d = a$
- $d \div a = b$
I tried many times to find a valid square or prove that it was impossible, but only got this:
- $a$ and $b$ can't both be positive, because then $c > b$, therefore $d < 0$, and $c \div d < 0$, making $a$ negative.
Is it possible / impossible? Please provide an example if it is possible or proof if it is not.
It is impossible. Your two first equations imply $d=-a$. The last one then shows $b=\frac{-a}{a}=-1$. The third one now reads $-a(a-1)=a$, and since $a$ is not $0$(for the last equation to make sense), we get $-a+1=1$, so $a=0$, contradiction.