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Gergonne’s trick

By on 01/02/2016

It was first analysed and generalised by the 19th century French mathematician Joseph Diaz Gergonne, and is described – including or not various extensions – in many recreational mathematics books, e. g. in Gardner’s “Mathematics, Magic and Mistery”. Here we present the basic version of the trick, using 27 cards.

The 27 cards are dealt face-open in three columns with 9 cards in each:

The performer asks a spectator to choose a card, but to name only its column. Say, the spectator has chosen the card the top card in the rightmost column in the picture above – he names the rightmost column as the chosen one. The performer collects the cards, and then deals them out again. In the described case, the cards will be rearranged as shown in the next picture:

The spectator is asked to name the column with “his” card again; in our example case, this would now be the leftmost column. The procedure “performer collects the cards, deals them out again, and asks the spectator for the column with his card” is repeated. In our case, the third layout of cards would look like this:

Now the spectator would again name the leftmost column. Once more the performer collects and deals the cards, but now he does not ask questions. Instead, he uses his mathemagical powers, concentrates – and reveals that the chosen card is now in the center of the layout:

In fact, the previous procedure will always bring the chosen card to the central position, provided the performer takes care of the following rules: always pick up the cards by columns, put the indicated column between the other two and deal the cards out by rows.

Why does it work? In this basic version, the secret is a basic principle of elimination: indicating the column reduces the number of possible cards to 9; by putting this column in the center and dealing out by rows, these 9 cards will be the central ones. A second indication of the column thus now reduces the number of possible cards to 3, which land in the central row after the second round of collecting and dealing. The final question is basically not needed – the card is now in the center of the indicated column, and the repeated procedure will bring it to the center.

We finish this short article by a simple question for the reader: If you start with 3n cards, how many repetitions of the procedure “collect-and-deal” are needed to bring the chosen card to the central position?