FunFact:
To "cross my heart" in russian "I'm giving you my tooth"
Дать зуб
‘Зуб даю!’ (I’m giving you my tooth!) By saying this, the speaker wants to show that they are so confident in their words that they are even ready to sacrifice a tooth.
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The idiom most likely originates from prison jargon. In prison, a person has nothing of value to use as a guarantee of their promise, but they can ‘give a tooth’, in other words, promise to pull out a tooth if what they say is not true.