Hidden Operation Rules

• The first number is $a$ and the second is $b$. Write them into the rule.
• Work out the result in BODMAS order.
• If the stem asks which pair gives a stated value, put each option's pair into the rule and keep the one that matches.
• A rule need not give the same answer both ways round: with $a \& b = ab - a$, $4 \& 6 = 20$ but $6 \& 4 = 18$.

Try the rules in this order on the first example:

• One operation: $a + b$, $a \times b$, $a - b$, $a \div b$.
• Squares and powers: $a^2 + b^2$, $(a + b)^2$, $a^b$, $a^2 + b$, $a^2 - b$.
• Mixed: $2(a + b)$, $2ab$, $ab + a + b$, $ab + a$, $ab + b$, $a(a + b)$, $(a + b) \times b$, $(a - b)^3$.
• A result next to a square or cube points to one: 50 is $7^2 + 1$, 343 is $7^3$.
• Test the rule on every example. A rule that fits one example is only a guess.
• When two rules fit every example, work out both. If they agree, the answer is safe. If they land on two different options, the question is ambiguous; prefer the rule that uses both numbers.

• Try the simple rules on each option in turn: +, −, ×, ÷, then the rules from the last concept.
• The rule must fit exactly three options. If it fits all four, or only two, try the next rule.
• Fast route: when all four results are the same number, ask what each pair of numbers does to reach it; the one that needs a different step is out.

• Write the step under each link of the first set: $\times 2$, $+ 10$, square, and so on.
• Confirm the same steps on the second set. If a step does not match, look for one that depends on the first number, such as ‘take away the first number’.
• Walk each option through the steps. The first wrong step rules it out; options are often built to pass the first step and fail a later one.
• Work on whole numbers only: SSC's note forbids splitting 13 into 1 and 3.