原文 Source：https://zhuanlan.zhihu.com/p/27185883

标题 Title: 扫雷图的结构以及影响难度的因素分析 Analysis on structure and factors affecting difficulty of minesweeper boards

作者 Author：张少武(Shao-Wu Zhang)

日期 Date: 2017-06-06

翻译 Translator: 濮天羿(Tian-Yi Pu)

• 扫雷图的结构——最被广泛应用的参数：3bv Minesweeper board structure - the most widely used stat - 3bv
• 3bv的计算方式 Calculating 3bv
• Op对扫雷图的难度影响 Openings affect difficulty
• Is对图难度的影响 Islands affect difficulty
  • 油水问题 The issue of premium
  • 移动距离问题 The issue of movement
• 关于Op和Is的一点相关性和杂谈 Relation between openings and islands and miscellaneous

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扫雷图的结构——最被广泛应用的参数：3bvMinesweeper board structure - the most widely used stat - 3bv

3bv是Bechtel's Board Benchmark Value的缩写，用来表示完成一张图所需的最少左键点击数。这是衡量一张扫雷图解决难度被普遍实用的参数，没有单位，就是一个数字，嗯。 3bv stands for Bechtel's Board Benchmark Value, meaning the least left clicks required to complete a board. It's the most popular stat among those evaluating the difficulty of boards. 3bv is just a number without unit, yes.

以一张初级图（标准初级图是8乘8的10雷图）为例： Taking a beginner board (8x8/10) as an example:

然而，雷界人眼中的图是这样的： However, the board looks like this in minesweepers' eyes:

1. Op（openings），即“开空”，简称“空”。用大众说法就是“一开一大片”的一块空白区域。其实这个空白区域就是一片数字0，只是0不显示而已（感谢雷友陈天力的精辟解释）。当然每个op的区域还包括空白区域边缘的一圈数字——被称为“Op边缘”，当op被打开的时候这些数字（图中黄线的部分）也是一起被打开的。Op的范围包括一块连续的空白区域及其周围的一圈数字（Op边缘），是不可分割的一个整体。显然图中有两个Op，并且每个Op算作一个3bv且只算做一个3bv！不管面积多大都是如此。 An op, or opening, is a blank region which opens at once. This blank region is in fact a cluster of invisible number 0's (thanks to Tian-Li Chen's brilliant explanation). Of course, an op includes the numbers, called op boundary highlighted by yellow outlines, surrounding zeros and getting uncovered with the op. The range of an op as a whole includes a connected blank region and its surrounding numbers (op boundary). Obviously the above board has 2 ops and each op is counted and only counted as a single 3bv, however big it is!

2. 雷（mines），这个没啥可说的。 Mines, not much to say.

3. 不属于Op边缘范围内的每一个数字都是一个3bv，即图中的黑框范围内的13个数字都是3bv。 Each number outside of op boundaries represents a 3bv, which means that the 13 numbers within the black outlines are all 3bv.

4. Is（islands），即“岛屿”。每个Is就是3bv组成的一个连续块。何为“连续”，就是哪怕两个作为3bv的数字方格之间哪怕只有角碰上了，也视作“连续”，如下图的红色Is。 An island, aka "is", is a connected chunk of 3bv. The connectivity includes the case where two 3bv numbers only share a corner, such as the red island below.

3bv的计算方式Calculating 3bv

**3bv = Op的个数+不属于Op边缘的所有数字的个数（即Is上的3bv总数），雷界扫雷以提高效率尽快解决3bv为指导思想。** 3bv = number of openings + number of numbers not on op boundaries (aka 3bv in islands). The basic principle of minesweeper is to solve 3bv as fast as possible.

依然以此图为例，很明显2个Op，红蓝黑三个Is上共有11个作为3bv的非Op边缘数字，所以这张初级的的3bv = 2+11 = 13。形容这张图的属性就可以说“这是一张2Op3Is的3bv = 13的初级图”，你也可以知道，只用左键完成这张图至少需要13次点击。 Taking this board again as the example. There are two openings and 11 non-op-boundary numbers in three islands outlined in red, blue and black, so this board has 2+11=13 3bv. A description of this board could be "this is a beginner board with 2op3is and 11bv". You may also know that at least 13 left clicks are required to complete this board.

Op对扫雷图的难度影响Openings affect difficulty

点开op边缘的数字并不能打开这个Op，直到你点到空白区域（即数字0）才能破开Op艰难地获得这个3bv。如下艰难过程： Uncovering cells on op boundaries will not uncover the op until you click in the blank region (or number 0) to get this 3bv finally. See below:

对的，我点了5次，只打开了一个Op只获得了一个3bv！因为四次点击都点在Op边缘(黄线位置)上，第五次才点到Op正体。 Yes, I clicked 5 times only to get one op which is only one 3bv! Because 4 clicks were on the boundary (outlined in yellow) before the 5th click in the interior.

Op这个3bv的获取难度较大，也许意识好或者运气好能一下子点中op正体，但大多数情况下你都需要2至6次点击才能“一开一大片”。对于雷界的人这往往是让人沮丧的，因为扫雷是以获取3bv的速度为主要数据，显然Op会大大拖慢获取3bv的进度。而非op的普通3bv点一个是一个，刀刀见血。 The 3bv of opening is hard to get. Maybe you are smart or lucky enough to get the op interior in one shot, but most of the times you will need 2-6 clicks to "open a collection of cells". This is often frustrating for minesweeper, since minesweeping aims at getting 3bv fast while openings are extremely slow to get. On the other side, every non-op 3bv is solved in one click which is much more efficient.

上图为3bv = 44的中级图，多达9个Op，我花了62次有效点击才解决，其中18次浪费在Op边缘上，扫了14.22秒。 The above is an intermediate board with 44 3bv and up to 9 openings. It costed me 62 effective clicks, 18 of which were wasted on op boundaries. The time score was 14.22s.

上图同样是3bv = 44的中级图，却只有1Op，没有一次有效点击浪费，只点了44次有效点击即解决44个3bv，只花了10.4秒。其实1Op这张图略难，但凭借图的高质量获得了不错的成绩。 This is also an intermediate board with 44 3bv, but only 1 opening. There was no wasted effective click and the board costed me only 44 effective clicks in 10.4s. This board is actually harder, but the high quality provided better scores.

结论：在不考虑图判雷难度的情况下，假设总3bv相等，Op数量越大，这张图的解决难度一般越大。但是我们在陷入困境的情况下希望“一开一大片”来提供新的判雷线索，未陷入困境的情况下又不希望点了半天只出一个Op浪费几次点击拖慢速度，人生不如意十之八九啊哈哈哈…… Conclusion: not considering the logical difficulty of a board, the difficulty tends to increase with openings. However, on one hand, when we get stuck, we wish to see an opening for information. On the other hand, when we break through easy parts, we hate to click many times only to get one opening. Life is hard, lol.

Is对图难度的影响Islands affect difficulty

油水问题The issue of premium

Is的数量就是3bv集合块的数量，显然这个数量越少，说明3bv块数越少（也就是3bv分布越集中），更有利于玩家快速解决。尤其对于标雷的玩家，大面积的Is会给双击带来更多的油水。例如下例，开局标雷之后…… The amount of islands is the amount of 3bv blocks. Less islands mean less blocks of 3bv (or more concentrated 3bv) and thus easier to solve fast. Large islands bring higher premium, especially for FL players. See below for an example after one flag:

一次双击the first chording

两次双击the second chording

三次双击the third chording

四次双击the fourth chording

这张图是个只有1Op1Is的好图，全部3bv集中为一整块。 This is a nice board with only 1 opening and 1 island. All 3bv form one piece.

在大块Is大油水的加持下，开局之后的1个标雷和4个双击获得了12个3bv！只用了五次点击就获取了左键12次点击的效果，这就是Is少、3bv集中对标雷玩法的利好。 With the help of the large island and high premium, the first flag and 4 chords earn 12 3bv! Getting the same effect as 12 left clicks in 5 clicks shows how less islands and more concentrated 3bv favor FL players.

移动距离问题The issue of movement

Is数量还会影响到移动距离，如图 The amount of islands also affect path, for example:

这张图是1Op1Is的3bv = 9的初级图，打开唯一一个op后，所有的3bv在右上角集中为一个Is，只需要在右上角处理完就可以了，移动距离很短很轻松。 This is a beginner board with 9bv and 1op1is. After uncovering the only opening, all remaining 3bv gather in the top right corner and we only need to solve that corner. The path is short and easy to move.

而反例如下：By contrast:

这张图是1Op4Is的3bv = 9的初级图，打开唯一一个op后，3bv分散在四角形成四个Is，需要到处移动,零敲碎打地解决，路径和操作难度大大增加，而且分散的3bv不利于连续点击加速，速度自然慢了很多。不要小看这几下移动，即使对于高手，拖慢个半秒不在话下。 This is a beginner board with also 9bv but 1op4is. After uncovering the only opening, the remaining 3bv scatter in four corners. We have to move everywhere and solve one or two 3bv at one time. Not only are the path and movement difficulty much higher, but also scattered 3bv oppose clicking fast. The solving speed is thus much slower. Don't underestimate these few movements. They can easily slow down by half a second even for skilled players.

结论：在不考虑图判雷难度的情况下，假设总3bv和Op数量相等，Is越多，这张图的解决难度一般越大。 Conclusion: not considering the logical difficulty of a board, the difficulty tends to increase with islands for fixed 3bv and op numbers.

关于Op和Is的一点相关性和杂谈Relation between openings and islands and miscellaneous

Op和Is的数量很大程度上是正相关的，毕竟Op对图是有切割作用的，Op越多，图被切得越零散，3bv趋向于越分散导致Is也变多。当然凡事无绝对，只能说整体趋势是这样。 The amount of openings and that of islands are quite relevant. Since openings have cutting effect on boards, more openings can cut the board into more pieces and introduce more islands. This is just an overall trend, not a strict rule.

Op和Is的增加会降低解决3bv的速度和效率，雷界一般认为不考虑判雷难度的情况下，它们越少越好。当然由于Op和Is对图的切割性，被切得越零散，图越容易扫开，因此它们反而还比较受新手玩家的欢迎哈哈哈……想要成为高手玩家，还是要克服判雷问题，让判断难度变成次要因素，才能更好的抓好图抓神图。 The increase of openings and islands can slow down speed and efficiency of solving 3bv, so it's widely accepted that less of them is better without considering logical difficulty. However, due to their cutting effect on boards, more openings and islands result in more scattered boards which are logically easier, so openings and islands are, on the contrary, favored by novice players. To become a top player, you have to overcome logic issues and make logical difficulty as a secondary issue so you can utilize nice boards and godly boards.