some & any

We use some in a positive sentences .

I'd like some cheese.

We use any in questions and negatives

Do you have any cheese?
We don't have any cheese.

But why tey use some here:

"Would you like some cake, then?"
"Would you like something to eat?"

We use SOME and ANY with plural nouns and uncountable nouns.

Some is generally used in positive sentences.

Any is generally used in negative sentences.

• I have some information for you about flights to Paris.
  (Positive - Uncountable)
• I don't have any information for you about flights to Paris.
  (Negative - Uncountable)

• We met some friends for drinks after work yesterday.
  (Positive - Plural Countable)
• I didn't see any friends there on Thursday.
  (Negative - Plural Countable)

• I think he will have some time to speak to you today.
  (Positive - Uncountable)
• I don't think he will have any time to speak to you today.
  (Negative - Uncountable)

You can also use SOME and ANY in a sentence without a noun if the meaning of the sentence is clear.

• I didn't eat any salad but Peter ate some. (salad)
• Sean took lots of photos of the mountains but Emma didn't take any. (photos)

Questions with Some and Any

Generally, we use ANY in questions.

• Do you know any famous people?
• Do you have any children?

But, SOME is used in the following circumstances:

1. When we are offering something.

• Would you like some coffee?
• Do you want some sugar for your coffee?

2. When we are asking for something.

• Could I have some salt, please?
• Can I have some fries with that?

3. When we are suggesting something.

• Why don't we rent some movies to watch tonight?
• Why don't you give her some advice?

You can also use SOME and ANY in a sentence without a noun if the meaning of the sentence is clear.

I didn't eat any salad but Peter ate some. (salad)

Sean took lots of photos of the mountains but Emma didn't take any. (photos)

Some & Any

We use "some" in a positive sentences.  ("a" = "one of...")

I'd like some cheese.

We use "any" in questions and negatives

Do you have any cheese?
We don't have any cheese.

But why do they use "some" here:

"Would you like some cake, then?"
"Would you like something to eat?"

Basically, the rule for some is not as strict as the rule for any.  Some also has a sense of reality about it (thus its use in positive sentences).

The suggestion behind "Would you like some cake?" is "I have some cake for you."  It is perfectly possible to replace some with any in both your sentences, but they sound far less definite, as if the cake/food must be searched for, or bought, first.