Missing Number in Grids

The order a strong student tries, on the first full row:

• Size check. Third number near first $+$ second: try the sum plus or minus a small number. Third number below the first: try a difference or a half. Third number far above both: go to products or powers.
• Common forms: $a + b \pm k$, $2a + b$, $a - 2b$, $a - \frac{b}{2}$, $b + \frac{a}{2}$.
• Prove it on the second row. A rule that fits one row only is a guess.
• Apply it to the last row, then find the answer among the options.
• Whole numbers only: SSC's note means 13 stays thirteen. Never split it into 1 and 3.
• A 3 by 3 square with no rule given: add each row, column and diagonal. In a magic square every line has the same sum.

• Common forms: $ab \pm k$, $k \times a \times b$, $\frac{a}{b} \times k$, $\frac{ka}{b}$.
• Divisibility hint: if $a$ divides exactly by $b$ in every row, start with $\frac{a}{b}$ and see what multiplier turns it into the third number.
• Product near the answer: work out $ab$ for two rows. If both miss the third number by the same amount, that amount is the $\pm k$.
• Check without dividing: if the rule is third $= \frac{ka}{b}$, then $b \times$ third $= k \times a$ in every row. Test each option this way.

• Know by heart: squares up to $20^2 = 400$ and cubes up to $10^3 = 1000$.
• Common forms: $a^2 + b^2$, $a^2 + b^3$, $a^3 - b$, $a^2 - b$, $(a + b)^2$.
• Find the square inside: take the nearest square below the third number away from it and see whether what is left is linked to the other number.
• Chain: the second number comes from the first (for example $\times 3 + 1$) and the third from the second (for example $\times 2 + 1$). Prove each step on every full row.