You have five numbers in a fixed order, four empty gaps and a target result. Insert each sign exactly once: +, −, × and ÷. One detail changes everything: perform the operations from left to right, without applying the usual precedence of multiplication and division. Try the 42 puzzles in the interactive Four Signs game.
Reading the chain correctly
In the first challenge the numbers are 5, 4, 3, 2, 1 and the target is 8. One solution is 5 − 4 + 3 × 2 ÷ 1: first 5 − 4 = 1, then 1 + 3 = 4, next 4 × 2 = 8, and finally 8 ÷ 1 = 8. Calculating 3 × 2 first would be a different problem.
A useful method
There are four signs and each is used once, so at most 4! = 24 arrangements need checking. Search thoughtfully: consider what division will do to the current result, whether subtraction takes it below zero, and work backwards from the target to constrain the last step. Harder challenges may pass through fractions and negative numbers; these are legitimate intermediate results, not mistakes.
Fractions without rounding
For example, 2 ÷ 4 − 6 + 8 × 10 gives 25: the successive results are 1/2, −11/2, 5/2 and 25. The game calculates each step as an exact fraction, so 1/3 is never replaced by an approximate decimal and a correct answer is not rejected because of rounding.
Source
The 42 puzzle configurations are adapted from “I 4 segni” in Numeri by Dario De Toffoli, Dario Zaccariotto and Silvia De Toffoli, Hachette, pp. 237–247. The interface, exact calculation checks and explanations are an original adaptation; the book scans are not reproduced.