The College Mathematics Journal | 2021
Fano, Galois, Hamming and a Card Trick
Abstract
Note that this trick uses the binary expansion of the secret number and hence you can find it by adding up the 2-powers in the upper left hand corners of the cards. However, the above trick does use one important assumption: that your friend tells the truth. Even if your friend lies one time, it fails. Can we modify the trick so that, even if the friend lies once, the number can be found? The answer is yes. To do this, we need more cards, but we can exploit an error-correcting code, originated by Richard Hamming [2]. The first nontrivial Hamming code is a code of length 7 that has 16 code words and can correct one error. This means that if your friend thinks about an integer between 0 and 15, is willing to answer 7 yes/no questions, and lies at most once, you should be able to determine which number he/she is thinking about. In the article [1] seven cards (and a base card) were presented to do this task. The computation was digital in the original meaning of the word. Each card had eight “digits” (tabs or fingers) and you had to find the number that was covered by at most one digit. This construction of the cards was improved in the article [3]. In the article [4] this computation was replaced by a mental computation. However what can we do to introduce a tool to help the performer to do this mental calculation? My suggestion is to use the Galois field GF(8) and the Fano plane. A field is an algebraic system where we can add elements, subtract elements, multiply elements and divide with nonzero elements. Well-known fields are the rational numbers Q, the