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This is an engaging activity in which students are given information and expected to make sense of it. It may lead to a discussion of modular arithmetic.
The video, or a live performance of the trick with a colleague, provides a hook to draw students into the problem.
Perform the trick three or four times, keeping a record of the four cards and the secret card. Ask them to discuss in pairs any ideas they might have about how the trick is done.
Share as a class any ideas that emerge and give students the chance to try out any suggestions with a pack of cards.
It is quite likely that the strategy used in the video won't emerge, so once students have appreciated the limitations of their suggested methods, hand out this worksheet. Give students time to make sense of the instructions and to perform the trick in pairs a few times. Ensure that they swap roles and have experience of both selecting the cards and 'guessing' the secret card.
"At the end of the lesson I am going to choose one of you at random and give you five cards. You will choose four cards to show to the rest of the class and I will expect everyone to be able to predict what the fifth card is!"
Finally you may want to discuss why the trick always works.
Four cards can be arranged in $4 \times 3 \times 2 \times 1 = 24$ ways. There are 52 cards in a pack. How can these 24 possibilities convey enough information to distinguish between 52 different cards?
There are some worked examples in the Hint.