Some equations are solved on the basis of a certain system. Find the correct…

Concept: in a "certain system" number-coding item the × sign is only a separator between the three given numbers, not an instruction to multiply. The solved lines are a code: one fixed rule turns the three numbers into the result, and a candidate rule may be used only after it reproduces every solved line exactly. Because each result is a two-digit number, the rule to look for is one that fixes a tens part from one of the numbers and then adjusts the units part with the others — that is, value = 10 × (first number) + (second number − third number).

Application: test that rule on the lines that are already solved, then apply it to the unsolved line.

1. 5 × 7 × 4 — tens part 10 × 5 = 50; units adjustment 7 − 4 = 3; 50 + 3 = 53, which is exactly the result given.

2. 7 × 5 × 8 — tens part 10 × 7 = 70; units adjustment 5 − 8 = −3; 70 − 3 = 67, which is exactly the result given.

3. 8 × 3 × 6 — tens part 10 × 8 = 80; units adjustment 3 − 6 = −3; 80 − 3 = 77.

Cross-check: the rule is exercised in both directions before it is used, because one solved line needs a positive adjustment (7 − 4 = +3) and the other a negative one (5 − 8 = −3); nothing new is assumed on the unsolved line. Working backwards agrees as well: 77 − 10 × 8 = −3, and 3 − 6 = −3.

Why the obvious readings are rejected:

• Ordinary multiplication fails on the very first solved line: 5 × 7 × 4 = 140, not 53.

• Adding the three numbers fails too: 5 + 7 + 4 = 16, not 53.

• Reversing the digits of the product of the first two numbers fits one solved line (5 × 7 = 35, reversed 53) but not the other (7 × 5 = 35, reversed 53, while 67 is required), so it is not the system in use.

Result: on this system, 8 × 3 × 6 = 77.