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Donald, from Verulam, had a good idea to try to ensure the message went through uncorrupted:
Repeat the original message after the true message and the check digit. For example, if 1110 is the original, and 11101 is the original and check, then 111011110 is the original and check and original again. Transmit this message. We can split this up, 1110 1 1110. If each number is changed, e.g. 1111 1 1110, then we look at the 1st original message '1111' and the 2nd original message '1110'. If they are the same, then this is probably the correct message. If not, then look at the check digit.
We've added this to our 'Toughnuts' to see if anyone can solve the rest of the problem. Can you be the first?