Doge, the king? (Faints)

Number Theory Level 4

Doge has become the king of a huge empire and wants to show the world that he is really intelligent. He is so over ambitious that his first action as king is to name the currency after himself: DogeCoin. He wants to issue \(d\) denominations of coins so that using no more than 3 coins, citizens can pay any amount from 1 DogeCoin to 36 DogeCoin in exact change.

Find the minimum value of \(d\) and all the corresponding denominations of DogeCoin. Submit your answer as the sum of these denominations.

Details and Assumptions:

  • As an explicit example, if you get the denominations as \( \{10, 12, 15\}\), then input the answer as \(10+12+15=37 \).

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