## Mind-Reading and Card Tricks: the Kruskal count

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Want another mind-reading trick? This one uses an ordinary deck of cards and is very impressive, although it only works about 85% of the time.

Here’s how to play. Have a person think of a number n between 1 and 10 and keep that number secret. You shuffle a deck of cards, and then slowly deal the deck face up card by card as the person watches. The nth card is a “key” card, in the words of Ivars Peterson, and the value of that card — counting aces as 1, face cards as 5, and all other cards as the number shown — tells the person how many cards ahead the next key card is. For example if the initial number is 7, then the 7th card is the first key card. If that card is a Jack (worth 5) then the person counts ahead (as you continue to slowly deal) 5 more cards, and that 5th card is the new key card. If that card is a 3, then the 3rd card after that is the next key card, etc. All of this is done in silence.

Eventually the deck runs out. You look at the person and, using your amazing mind-reading/mathematical powers, you show them their final key card!

The way you figure this out is that, while you’re slowing dealing out the deck, you are doing the same kind of counting! You silently start with your own number — it helps to start with a small number, like 1 or 2 — and use that to obtain your own key card. Partway through the deck it is likely that both of you will end up on the same key card, and from then on all your key cards will be the same. When this happens, your final key card is the same as the other person’s, so it’s your own final key card that you pull out and show. You can increase the likelihood of this trick working by using more cards (e.g. two decks shuffled together) or by assigning a smaller value to face cards.

This phenomenon is known as the Kruskal count, named after Martin David Kruskal. It applies to card tricks like this, and can also be used to predict the last word that someone will end up on in a passage (in this case the person chooses a word near in the first line or two of a passage, counts how many letters are in it, moves forward that many words, counts those letters, moves forward by that number of words, etc. You predict the last word they obtain in the passage).

Ivars Peterson has written nicely about the Kruskal count in Math Trek and (with pictures) in Science News for Kids. There is also a nice example in which it’s applied to The Wizard of Oz in Computer Science For Fun.

A memorial page to Martin Kruskal , who passed away last year, is hosted here by Rutgers University.

### 3 Responses to “Mind-Reading and Card Tricks: the Kruskal count”

1. TwoPi Says:

I did a variant of this card trick in class yesterday: after shuffling the deck, I muttered that I wasn’t sure there were 52 cards in the deck, so I dealt them out on the desk. [As I did this, I did a Kruskal count in my head.] Now knowing the likely final key card, I wrote down the prediction on a slip of paper and put it in an envelope. At that point I asked for two volunteers (one to do the count, one to hold the envelope), and the rest went as you’d expect. (Quite well, that is.)

2. Bryan Says:

A variant of this is to assign face cards as 1 instead of 5.

3. Magical Results and P=NP « Gödel’s Lost Letter and P=NP Says:

[…] trick uses an ordinary deck of cards. Each card has a value: an ace is , all picture cards are , and the […]