Deck of Hearts - an old school problem

Logic Level 2

A math teacher holds a pre-organized deck of cards in her hand - with all the cards faced down. This pre-organized deck of cards (total # of cards = 13) only contain the cards of heart .

The teacher begins to draw from the top of the deck:

1st card is put on the table.
2nd card is put back into the bottom of the deck of cards.
3rd card is put on the table.
4th card is put back into the bottom of the deck of cards.
5th card is put on the table.
...........
22th card is put back into the bottom of the deck of cards.
23th card is put on the table.
24th card is put on the table.

The sequence of cards that were put on the table was:

Q J K 3 5 7 10 9 4 8 2 A 6

Question: In which order did the teacher pre-organize the cards in order for this sequence to appear on the table?

Hint: Experiment in real-time with a deck of cards containing only the cards of heart.

Q A J 9 K 2 3 4 5 6 7 8 10 Q J K 3 5 7 10 9 4 8 2 A 6 6 2 9 7 3 J Q K 5 10 4 8 A 6 A 2 8 4 9 10 7 5 3 K J Q Q 6 J A 2 K 3 8 5 4 7 9 10 Q K 5 7 9 4 8 A 6 2 10 3 J 2 3 4 5 6 7 8 9 10 J Q K A

2 solutions

Joshua Lowrance
May 15, 2018

I started with the arrangement of cards already on the table, and worked backwards from there. Take the 6 from the table (step 24), then take the A from the table and put it on top of the 6 (step 23). From then on, you will follow the pattern of 1) take the bottom card from the deck you have in your hand and put it on the top of the deck (this is the reverse of taking the card on the top and putting it on the bottom) then 2) take a card from the table and put it on the top of the deck in your hand (this is the reverse of taking the card on the top and putting it on the table). Do this until you put the Q on the top of the deck. You should have Q A J 9 K 2 3 4 5 6 7 8 10

Jonathan Helsing
May 13, 2018

This problem was introduced to me by my father and, back then, the sequence of cards that were put on the table was: 2 3 4 5 6 7 8 9 10 J Q K A, which gave the problem more of magic trick feeling. There are probably several ways of solving this problem. The first time I solved it, I used a deck of cards and I peeked - so lets call that the "peeking solution".

PEEKING SOLUTION:

One thing that remains constant in this problem is the pattern/algorithm/mechanic of the game, and we will obtain it by peeking.

My pre-organized deck were: 10 J 6 A 5 8 7 Q K 3 2 4 9

The sequence of cards that appeared on the table was: 10 6 5 7 K 2 9 A Q 4 8 J 3

Just to make it a little more simple to understand, we replace the numbers with variables: a = 10, b = J, c = 6, d = A, e = 5, f = 8, g = 7, h = Q, i = K, j = 3, k = 2, l = 4, m = 9

The sequence of cards that appeared on the table, using variables, was: a c e g i k m d h l f b j

a b c d e f g h i j k l m <==> a c e g i k m d h l f b j

With above "rule" you can take any PRE or POST sequence of cards and translate it in both directions. To solve this problem for example:

Q A J 9 K 2 3 4 5 6 7 8 10 <==> Q J K 3 5 7 10 9 4 8 2 A 6

Shouldn't the new sequence start as "QK" instead of "QA?"

Blan Morrison - 3 years ago

Deck of Hearts - an old school problem

2 solutions

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