Problem
Suppose the nation of Bitcoinia decides to convert its national lottery to use Bitcoin. A trusted scratch-off ticket printing factory exists and will not keep records of any values printed. Bitcoinia proposes a simple design: a weekly public address is published that holds the jackpot. This allows everyone to verify that the jackpot exists, namely there is a UTXO bound to that address that holds the jackpot amount. Then a weekly run of tickets is printed so that the winning lottery ticket contains the correct private key under the scratch material.
1. If the winner finds the ticket on Monday and immediately claims the jackpot, this will be bad for sales because players will all realize the lottery has been won. Modify the design to use one of the locktime opcodes to ensure that the prize can only be claimed at the end of the week. (of course, you cannot prevent the winner from proving ownership of the correct private key outside of Bitcoin). The Bitcoin script opcodes are listed here. Alternatively, explain how to use a 2-out-of-2 multisig for this.
2. Some tickets inevitably get lost or destroyed. Modify the design to roll forward so that any unclaimed jackpot from Week n can be claimed by the winner in Week n+1. If the Week n+1 jackpot is unclaimed, then the jackpot from both weeks n and n + 1 can be claimed by the winner of Week n + 2, and so on. Can you propose a design that works without introducing new ways for the lottery administrators to embezzle funds beyond what is possible on the basic scheme described at the beginning of the question? You may assume that the lottery system runs for 1000 weeks, and then shuts down permanently. Consider using multisig.