In the pattern shown, A,B , C, and D are prime numbers and all eight numbers are different. The sum along any 3-number row or column is the value of S. What is the smallest possible value of S?
Hi All, So here is my problem, I did some thinking and this is what I got:
17+A+B=S
7+C+D=S
So on, and so on. I tried to replace numbers, but I was left with one equation, 3 variables. Is it possible to receive some help for this?
$A+C = B+D + 2\\ A-B+C-D = 2\\ A+B-C-D = -10\\ 2A-2D = -8\\ D-A = 4\\ C-B = 6$
So now we go through the list of prime numbers looking for a gap of 6 and a gap of 4.
$(7,3)$ is the smallest pair with a gap of 4, but we can't use 7 (because it has already been used). So, now we are looking at primes greater than 17.
The smallest pairs with a gap of 4 are $(23,19)$ and $(41,37)$ and the smallest pairs with a gap of 6 are $(29,23)$ and $(37,31)$
$(23,19),(37,31)$ minimize $S$
\begin{array}{}&&13\\17&19&31\\&37&23&7\\&11\end{array}