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This Venn diagram shows the relationship between the check bits and the message bits. We can use this to work out which digit has been altered. Suppose we find that yacd and zbcd contain an odd number of 1s, and xabd contains an even number of 1s. We look at the intersection of y, z and not-x (any region that x isn't in), which is c. This therefore tells us that c has been altered in transmission.
Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?
Here is the start of a six-part challenge. Can you get to the end and crack the final message?
Can you work out what size grid you need to read our secret message?