I have come across the two following sentences:

> 遊 **ぶ** ことは犯罪です | Playing is a crime
>
> 遊 **び** は終わりだ。 | Fun and games are over (lit. playing is ending)

Why is `遊ぶ` in plain/dictionary form in the first, and then ます form in the
second? Also what are the difference when they are in each of these forms?

This question goes with any verb.

Thanks.
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grammar verbs form
When to use verb ます vs plain form
First, 遊び is actually a noun, not a verb. It is formed from the verbal root.
There is a group of nouns in Japanese which is derived from the root and uses
this form (continuative form) as a standalone noun. It seems more common in
the godan verb category. Some common examples are:

> 休み (a rest) from 休む (rest)  
>  違い (a difference, a discrepancy) from 違う (differ)  
>  育ち (upbringing, growth) from 育つ (grow)  
>  狙い (an aim) from 狙う (aim)  
>  泳ぎ (swimming) from 泳ぐ (swim)

As for the difference between plain forms and masu forms, masu forms are
usually used in situations which require a degree of politeness or formality.
They are typically used between people who do not know each either well or in
situations which require formality. This is not an exhaustive list of when to
use that form.

Explaining the spectrum of uses for the plain form would require a very in-
depth answer, but in general, it is used in casual forms of Japanese, in
relative clauses, in modifying phrases, with nominalizers (as in your
example), among others.

Finally, as to the question of the difference between using 遊び and 遊ぶこと, this
is actually quite a philosophical question. In language, when do you use a
verbal phrase instead of noun phrase if both are possible? One answer is that
it is somewhat of a stylistic difference. You could argue that there may be
semantic differences between the two, but it's not always easy to distinguish.
For example, in English, would you say "playing video games is fun" or "gaming
is fun"? Are they different? Why choose one over the other? Likewise, someone
might prefer to use 遊ぶこと if they want to emphasize the action of the verb, or
vice versa if they want to stress the 'thing' that is 遊び. Having said that,
they are not interchangeable because of how they interact syntactically with
other phrases. For example, the sentence 計算に違いがある is acceptable but 計算に違うことがある
is unnatural (when intending the meaning "there is a discrepancy in the
calculation"). So, in my opinion, the choice between 遊ぶことor 遊び is based on
either stylistic or syntactic considerations, possibly on semantic
considerations if there are clear differences between the phrases.
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As others have pointed out, please don't refer to 遊び as "ます form" - it's not
that if it doesn't actually end in ます.

Both 遊び and 遊ぶこと/遊ぶの are noun forms of 遊ぶ, and in most cases both can even be
translated as "playing".

I have never been taught this, but the "feel" I have for things like 遊び versus
遊ぶこと, is that the former speaks more often to what results from playing - or
the "thing you play" - as opposed to the act itself. This idea may or not be
useful - it is definitely not always true.

According to this heuristic, 考え is the thought(s) I have,

  * 考えること is "to think". 遊び is the games I play or the fun I have, while 遊ぶこと is "to play" or "playing".
  * 終わり and (お)しまい are ends or "the end"; 終わること and しまうこと are "to end" or "ending".
  * 感じ is "my/your feeling", 感じること is the act of "feeling".
  * 20キロ走り is "a 20-kilo run", 20キロ走ること is "running 20 kilos"

Another way of thinking about it is, "can I put an article (a/an/the) before
it"? The masu-stem ones, generally you can; the こと ones, generally you don't.

But as I said, this idea also doesn't hold everywhere. 作り isn't "a creation".
You don't use 描き for "a drawing". Food isn't 食べ, 食べ物 is.

Also note that this stem form doesn't always indicate a noun. It's also used
to combine verbs, like 走り回る, "to run around". And sometimes it just means
"and" (mostly in the written language): 山本博士は、知恵があり, 素敵な人間なの. "Professor
Yamamoto is a fine specimen of humanity, brimming with wisdom."
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