Tanya Khovanova's Math Blog

Conway’s Absent King Trick

27th July 2024, 02:21 pm

The fine tenacious boys nicely joke to hated servant girls sick for (absent) king.

The picture shows John Conway’s notes in my journal. They are the mnemonic for setting up his trick. For example, the word “tenacious” sounds similar to “ten” and “ace”. Hence, we arrange 13 cards in this order: 3, 5, 10, A, J, 9, joker, 2, 8, 7, Q, 6, and 4. We also put the king aside. The trick looks better if the cards are the same suit.

The fun part of the trick is the story he told while showing it. Unfortunately, I do not remember the story. My only other note says:

One, two, three are done by me. Four, five, six: they do the tricks. Seven, eight, play them straight. We all try nine for quite a long time. The king is back.

Here is my attempt to recover the trick. We arrange the cards in the above order. We keep the cards face down so that the three is on top. Now, we spell the word “ACE”, and for each letter, we move one card from the top to the bottom of the pile. Then, we flip the next card from the top of the pile, and “tada”, the card is an ace. We put it aside. Now, we repeat the process by spelling “TWO”, and the next card after that is a two. We do the same for “THREE”.

But, when we try the same process for “FOUR”, we get the joker instead. This is not surprising if you remember that “Four, five, six: they do the tricks”. We put the joker at the bottom of the pile and continue. In the next round, after spelling “FOUR” again, we get 4, which we put aside. We proceed by spelling “FIVE” and getting a joker, then getting 5 after the second spelling. The same happens with “SIX”. Then we continue with “SEVEN” and “EIGHT” without getting the joker.

Then, we try “NINE”, and get the joker. Then, we spell it again and again and keep getting the joker. Clearly, we are in a cycle that can go on forever. If you recall the quote, “We all try nine for quite a long time.” To get out of this cycle, we remember our king and put it on top of the pile when the joker is on the bottom. We start again. And now we can spell “NINE” and get 9. We are back to normal with “TEN”, “JACK”, and “QUEEN”, too. The king, however, appears on the second try after spelling “KING”, getting the joker, and spelling “KING” again.

I do not remember the details of John’s performance. I tried to find the trick online but only saw it briefly mentioned in Mathematics, Magic, and Mischief with John H. Conway.

Have you seen this trick? Any juicy details are welcome in the comments.