Problem 93

Arithmetic Expressions

By using each of the digits from the set, ${1, 2, 3, 4}$ , exactly once, and making use of the four arithmetic operations $(+, -, \times, /)$ and brackets/parentheses, it is possible to form different positive integer targets.

For example,
$\begin{aligned} 8 & = (4 \times (1 + 3)) / 2 \\ 14 & = 4 \times (3 + 1 / 2) \\ 19 & = 4 \times (2 + 3) - 1 \\ 36 & = 3 \times 4 \times (2 + 1) \end{aligned}$
Note that concatenations of the digits, like $12 + 34$ , are not allowed.

Using the set, ${1, 2, 3, 4}$ , it is possible to obtain thirty-one different target numbers of which $36$ is the maximum, and each of the numbers $1$ to $28$ can be obtained before encountering the first non-expressible number.

Find the set of four distinct digits, $a < b < c < d$ , for which the longest set of consecutive positive integers, $1$ to $n$ , can be obtained, giving your answer as a string: $a b c d$ .

算术表达式

使用集合 ${1, 2, 3, 4}$ 中每个数字恰好一次，再加上四则运算 $(+, -, \times, /)$ 和括号，可以表示许多不同的整数。

例如：
$\begin{aligned} 8 & = (4 \times (1 + 3)) / 2 \\ 14 & = 4 \times (3 + 1 / 2) \\ 19 & = 4 \times (2 + 3) - 1 \\ 36 & = 3 \times 4 \times (2 + 1) \end{aligned}$
注意不允许把数字连起来使用，比如 $12 + 34$ 。

使用集合 ${1, 2, 3, 4}$ ，可以表示出 $31$ 个不同的数，其中最大值是 $36$ ，并且可以连续表示 $1$ 至 $28$ 之间的所有数。

考虑所有四个不同数字 $a < b < c < d$ 构成的集合，求其中可以从 $1$ 开始连续表示最多个正整数的集合，并以字符串 $a b c d$ 的形式给出你的答案。