For the sum scoring rule:
2^(3!*2 − 1) − 3^3 = 2021 (14)
For the product scoring rule, I’m not sure I can do better than:
√(2^(4! − 2)) − 3^3 = 2021 (144)
Puzzle 2
Roll d1811 and d1907 for a number between 1 and 3453577. This upper bound is only 1 more than a multiple of 2022, so there’s only a 1⁄3453577 chance that a reroll will be needed. This results in an expected 2.0000005791 rolls.
Puzzle 3
This solution uses two coins, one with [1/2], and one with [1/2021].
Flipping [1/2] eleven times can simulate a d2048. If the result is in 1-2021, then done. Otherwise, subtract 2021 to get a result in 1-27. Note that this result can be used to simulate a d3, d9, or d27.
Combining the previous result with seven more [1/2] flips can simulate a d3456. If the result is in 1-2021, then done. Otherwise, subtract 2021 to get a result in 1-1435. Note that this result can be used to simulate a d5, d7, d35, d41, d205, d287, or d1435.
Repeating the above process allows us to simulate, among others, a d2, d3, d5, d7, d31, or d101. These can then be combined to simulate any of the coins from the example solution except the [1/2021], which we already have.
The actual solution involves pre-computing the maximum number of each type of coin flip that could be needed to complete the example solution. Any time a result in 1-2021 is achieved, the process terminates immediately with that result. If all necessary coin flip results have been computed without terminating early, then proceed with the example solution using the pre-computed results as required in the order they were computed.
If there is a way to simulate a [1/2021] coin with a [1/2] coin, then only one coin is needed.
Puzzle 1
For the sum scoring rule:
2^(3!*2 − 1) − 3^3 = 2021 (14)
For the product scoring rule, I’m not sure I can do better than:
√(2^(4! − 2)) − 3^3 = 2021 (144)
Puzzle 2
Roll d1811 and d1907 for a number between 1 and 3453577. This upper bound is only 1 more than a multiple of 2022, so there’s only a 1⁄3453577 chance that a reroll will be needed. This results in an expected 2.0000005791 rolls.
Puzzle 3
This solution uses two coins, one with [1/2], and one with [1/2021].
Flipping [1/2] eleven times can simulate a d2048. If the result is in 1-2021, then done. Otherwise, subtract 2021 to get a result in 1-27. Note that this result can be used to simulate a d3, d9, or d27.
Combining the previous result with seven more [1/2] flips can simulate a d3456. If the result is in 1-2021, then done. Otherwise, subtract 2021 to get a result in 1-1435. Note that this result can be used to simulate a d5, d7, d35, d41, d205, d287, or d1435.
Repeating the above process allows us to simulate, among others, a d2, d3, d5, d7, d31, or d101. These can then be combined to simulate any of the coins from the example solution except the [1/2021], which we already have.
The actual solution involves pre-computing the maximum number of each type of coin flip that could be needed to complete the example solution. Any time a result in 1-2021 is achieved, the process terminates immediately with that result. If all necessary coin flip results have been computed without terminating early, then proceed with the example solution using the pre-computed results as required in the order they were computed.
If there is a way to simulate a [1/2021] coin with a [1/2] coin, then only one coin is needed.
Puzzle 1 does not allow the use of 1 (also, I changed the scoring to the product rather than the sum)
Updated and added solutions for Puzzle 2 and Puzzle 3.