It was entitled Solvable and Unsolvable Problems, and first gave an example of a 'solvable' problem. This was a solitaire game (actually the 'fifteen puzzle') in which, as he described, there were only a finite number of possibilities to consider (namely 16! = 20,922,789,888,000). Hence, in principle, the game could be 'solved' simply by listing all the possible positions.

这篇文章题为《可解问题与不可解问题》。图灵首先给出了一个可解问题的例子，这是一个纸牌解谜游戏，具有有限种可能的情况（即16！= 20, 922, 789, 888, 000），理论上可以穷举所有的情况，所以这个游戏是可解的。

This helped to illustrate the nature of an absolutely 'unsolvable' problem, such as he went on to describe, but the large number also demonstrated the gap between theoretical and practical 'solvability'.

通过理解这个问题，有助于理解他接下来给出的"不可解问题"。然而，这个巨大的数字，却体现了"可解"在理论和实践上的不同意义。

As it happened, of course, the Bombe had indeed exploited the finiteness of the Enigma by just such a brute force method, but in general the knowledge that a number is 'only' finite is not of practical significance.

图灵当年的炸弹机，确实利用了谜机的这种有限性，但是一般而言，这种有限性并不总是有用的。

One cannot play chess, nor deduce all the wirings of an Enigma machine, by knowing that the possibilities are finite.

一个人不可能通过穷举来下棋，同样也不可能通过穷举解出谜机的配线。

The 'fifteen puzzle', indeed, poses a tough problem to the computer programmer. Turing machines, when embodied in the physical world, are severely limited by considerations other than those of logic.

事实上，即使对于计算机程序来说，这个纸牌游戏也是一项非常艰巨的任务。在逻辑的世界里，图灵机享有很大的自由，但他一旦进入物理世界，就会受到各种因素的严重限制。

While some physical quantities (such as temperature) may be described by one number, in general they will require a set of numbers; anything like a direction in space, for instance, will do so.

虽然有些物理量（比如温度）可以表示成一个数，但更一般的情况下，会表示成一个集合，比如一个物体的空间位置。

It is usual to 'index' this set by a letter of the alphabet. From a modern point of view the structure of the set is a reflection of the group of symmetries associated with the physical entity, and it is common to use a different type of letter (e.g. lower-case, upper-case, Greek) when different symmetry groups are implied.

人们通常会用字母来索引这样的集合。这些集合的结构，反映出一种关于实体的对称群，当指称不同的对称群时，人们会使用不同类型的字母（大写、小写或希腊字母）。

The word 'fount' made this principle explicit.

"活字"这个词，直观地体现了这个理论。