Maths prize puzzle 27 January 2017: the 21 card trick

21cards

You choose one card out of 21 without telling the magician which one.

The 21 cards are dealt out face up in three columns of 7 cards each. You tell the magician which column contains your card. The magician picks up the cards, column by column, so that the column with your card is in the middle.

She or he deals them out across three columns again, top card in first column, 2nd card in 2nd column, 3rd card in 3rd column, 4th card in 1st column, etc. Then the same process (tell which column, pick up, deal again). Then you tell which column a third time. The magician picks up and identifies your card as the 11th card.

Explain mathematically how this is done. See if you can generalise to different numbers of cards.

Bolaji Atanda and Mohaned al-Bassam solved the puzzle and won prizes.

The 21 card trick is explained with pictures at:

http://blog.themathmom.com/2010/01/card-tricks.html

You can do it in m repeats of the point-to-column process with Nm cards in N columns. Simplify by having the magician put the selected column’s cards at the top when picking up and dealing them across the columns first.

After the first point to column/ pick up/ deal process the chosen card is within the first Nm−1 cards, i.e. within the first Nm−2 rows.

After the second similar process, within the first Nm−3 rows.

After the (m−1)’th similar process, within the first row (with the first N0 rows).

Now you point to the column, and the card is the first card of that column.

So you can do it in 3 repeats (as with the 21 card trick) with 27 cards in 3 columns, or 125 cards in 5 columns, etc.