"It develops games to study math" - "to" is part of the infinitive form. "It develops games to studying math" - to me, it doesn't sound correct. We can use gerund after preposition "to", when "to" is: a) part of a phrasal verb or some phrasal combination (e.g. I look forward to meeting you), b) part of a combination with an adjective (e.g. be addicted to doing sth) c) part of a combination with a noun (e.g. addiction to gambling). In your sentence I would use preposition "for" - It develops games for studying math".

You can say either: It develops games to study math It develops games for studying math

"It develops games to study math" - "to" is part of the infinitive form. "It develops games to studying math" - to me, it doesn't sound correct. We can use gerund after preposition "to", when "to" is: a) part of a phrasal verb or some phrasal combination (e.g. I look forward to meeting you), b) part of a combination with an adjective (e.g. be addicted to doing sth) c) part of a combination with a noun (e.g. addiction to gambling). In your sentence I would use preposition "for" - It develops games for studying math".