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From what I infer from searching online, 解く【とく】refers to "untying" or solving abstract concepts such as math problems. 解く【ほどく】, on the other hand, relates to untying physical objects like one's shoelaces. If this assumption is correct, in the case of this example 「くっついているものを解き分ける」, the correct 解く to use should be ほどく, right?

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[解く]{とく} and [解く]{ほどく} overlap in meaning for untying/undoing things that are tied, wrapped, or tangled. These "things" can be physical objects (like a rope tied in a knot), but they can also be abstract (感情のもつれ, i.e. tangled feelings/emotions).

For the example of untying one's shoelaces, you could use either [解く]{とく} or [解く]{ほどく}. The difference between the two, according to 明鏡国語辞典, is that [解く]{ほどく} is comparatively more of a spoken language/colloquial expression (emphasis mine in the following excerpt):

ほど・く【解く】
〘他五〙結んであるもの、縫ってあるもの、もつれたものなどをときはなす。とく。
「ひもの結び目を━」
「靴のひもを━」
「荷物を━」
「着物を━・いて縫い直す」
「からんだ釣り糸を━」
(表現)「[解く]{とく}」に比べて、口頭語的な言い方。

Another more obvious difference between the two, as you inferred, is that [解く]{とく} has a wider range of possible meanings, e.g. "solving" puzzles/problems, "lifting" a ban or restriction, or "clearing up" a misunderstanding.

As for your provided example 「くっついているものを解き分ける」, I believe it should read [解き分ける]{ときわける} as that sounds more natural to me, and there is this weblio entry that confirms this reading, but I would wait for a native speaker's input on this one.

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