Read the equation before choosing a number

An equation with several operations is a set of instructions. Before trying a digit, identify which parts belong together. This prevents a correct calculation of the wrong expression.

The order that Mathoji uses

Work inside brackets first. Apply square roots or cube roots to their operands. Multiplication and division then share the next priority, followed by addition and subtraction. Operations at the same priority are read from left to right. A × symbol means multiplication; ÷ means division.

For example, 2 + 3 × 4 is fourteen: multiply three by four, then add two. But (2 + 3) × 4 is twenty because the brackets change the order. Neither expression is evaluated by simply doing every operation from left to right.

Use the final blank as a remainder

In _ × _ + _ = 34, the multiplication comes first. Calling the blanks a, b and c gives ab + c = 34. Because the final digit is between zero and nine, the product must be between twenty-five and thirty-four. You can now search a short section of the multiplication table instead of trying a thousand ordered triples.

The products 25, 27, 28, 30 and 32 work. Calculate the remaining digit by subtraction. For instance, 4 × 8 makes thirty-two, so the final blank must be two. Reversing the multiplication to 8 × 4 repeats that same solution.

Division needs two checks

A divisor must never be zero. In the standard Easy through Expert puzzles, division must also be exact at the point where it is performed. A fractional intermediate result is not rescued by a later multiplication. Master puzzles explicitly allow fractional arithmetic and show their available expression keys.

Try the rule

What is 8 ÷ 2 × 3? Think about the equal priority of division and multiplication before opening the answer.

When a guess fails, check the order before replacing its numbers. A useful habit is to calculate each bracket or product separately, then combine the results in the original order.