The difference between 「가든지 말든지」 and 「얼마나 춥던지」

When the sentence means you may pick any one of two or more things, as in 「치킨을 시키든지 피자를 시키든지 네 마음대로 해」 (order chicken or order pizza, whatever you like), you use -든지. The same goes for sentences where the result stays the same whichever way things turn out, as in 「싫든지 좋든지 따를 수밖에 없다」 (like it or not, you have no choice but to follow). It also fits sentences saying anything is fine, such as 「무엇을 그리든지 잘만 그려라」 (draw whatever you like, just draw it well). It can be shortened to -든, and -든가, as in 「가든가 말든가」, is a standard form with the same meaning.

So 「먹든지 말든지」 (eat it or not) takes 든지 on both sides. Writing only the second half as 「먹든지 말던지」 sounds off, and the National Institute of Korean Language advises using the same ending on both sides in such sentences.

When you recall something in the past and connect it to what follows, as in 「얼마나 춥던지 손이 다 곱았다」 (it was so cold that my hands went stiff), you use -던지. The National Institute of Korean Language describes it as an ending that links a vague question about a past matter to the fact in the next clause. Put simply, it feels like saying, "I can't tell you how cold it was, so my hands stiffened." It often shows up when expressing surprise or worry, as in 「아이가 얼마나 밥을 많이 먹던지 배탈 날까 걱정이 되었다」 (the child ate so much that I worried about a stomachache).

In 「동생도 놀이가 재미있었던지 더 이상 엄마를 찾지 않았다」 (maybe my younger sibling also enjoyed the game, since they stopped asking for mom), the one attached after -었- is the same ending. Because it looks back on something already over, using 든지 in 「얼마나 놀랐든지 몰라」 (I was so startled) is wrong.

Just ask whether it is a choosing word or a recalling word. If you can line up two options in the pattern 「~든지 ~든지」, it is 든지. If the sentence carries 「얼마나」 (how much) and looks back on something that already happened, it is usually 던지.

This distinction is in Article 56 of the Hangeul Spelling Rules: endings that refer to past events are written -더라 and -던, while the sense of choosing or not caring is written -든. In speech the two sound almost the same, so the surest way is to pause for a moment when writing and think about the meaning.