Question

Using the digits 0-9, no more than once, complete the puzzle so that the sum of each side is equivalent.

Answer 1

`\begin{matrix}8&2&7\\ 4&&1\\ 5&3&9\end{matrix}`

Explanation:

`\begin{matrix}a&e&b\\ f&&g\\ c&h&d\end{matrix}`

From the table,we can extract these equations:

a + e + b = a + c + f = b + g + d = c + h + d

a + e + b = a + c + f ⇒ e + b = c + f

b + g + d = c + h + d ⇒ b + g = c + h

`\begin{Bmatrix}e+b=c+f\\ b+g=c+h\end{Bmatrix} \Longrightarrow g-e=h-f\Longrightarrow g+f=h+e`

2 + 3 = 4 + 1 then let’s consider :

e = 2 ; h = 3 ; f = 4 ; g = 1 ,which satisfies g + f = h + e

The table becomes:

`\begin{matrix}a&2&b\\ 4&&1\\ c&3&d\end{matrix}`

From the equation b + e = c + f we get b + 2 = c + 4 then b = c + 2

If we consider c = 5 ⇒ b = c + 2 = 5 + 2 = 7

Then the table becomes

`\begin{matrix}a&2&7\\ 4&&1\\ 5&3&d\end{matrix}`

Then a + 9 = d + 8 ⇒ d = a + 1

If we consider a = 8 ⇒ d = 9

`\begin{matrix}8&2&7\\ 4&&1\\ 5&3&9\end{matrix}`

I know it’s not perfect reasoning ,but it may help.