Kruskal Chains

 

For an introduction to the Kruskal chain, watch this video by CountVsauce2: (1853) The Number Illusion You Won't Believe - YouTube. The video shows a popular magic trick using the hours of a clock and is based on a Kruskal chain. The host explains how the trick works and once you understand how it works, the principal can be applied to other situations.

Here is how to expand the Kruskal chain to a card trick. Shuffle a normal 52-card deck of playing cards. Deal them out face up into 4 rows of 10 cards and a 5^th row of 12 cards.  Ask your audience member to pick one card from the top row but not to reveal it to you. Instruct them to count ahead in the matrix of cards based on the number of the card (aces = 1 and face cards = 5). When they land on the next card, they repeat the procedure based on the value of the card they land on, and they keep counting until they cannot proceed any further. As your participant is choosing the starting card, you need to pick any starting card and complete the exercise to the last row. Before your participant begins the counting process, announce you know which card they will be their final landing card – this will be the last card you landed on. Based on the Kruskal chain, most starting cards will have “chains” that intersect and once they do, the remainder of their sequences will be the same. There are some instances, where there will be one or more starting cards that produce a sequence that doesn’t intersect the other chains. This will occur about 15% of the time. If one uses two decks and therefore allowing for more chances of chains intersecting, the trick will fail to work only 5% of the time.

Update 4/9/2023

Interesting note - if one takes a new, unshuffled deck and deals it out in order as described above, the ending point will be the third card from the end of the last row - the three of spades.

Update 8/24/2024 - another post feature a Kruskal count has been added to this blog and it will become a proposed sequence for the OEIS.