Well,

Without any preface, I am just coming to the point.

**Problem : **You have standard deck of 52 cards. You are allowed to choose 5 cards randomly. Now, you hand over four cards one after another to your partner. The problem is that your partner should be able to guess the fifth card remaining in your hand correctly. What strategy would you devise to communicate the fifth card to your partner? Please note that you can not use orientation of the card ( handing it over to your partner with 90 degree tilt ) or any facial expression or any gestures. The only things that you have to use for communication are 1) choosing which four cards to hand over 2) order in which you hand them over to your partner.

**Solution :**

I saw this puzzle at a couple of place and I was wondering wow, it must be tricky to use only 4 cards and locate a card out of 48 possibilities. But as I put my mind to it, this puzzle seemed extremely simple to me. Here is the strategy that you and your partner can follow.

Note that, because you have chosen 5 cards, there must be at least two cards of the same suite because of pigeon hole principle. This way definitely you can communicate in which suite the fifth card belongs. Now let us have an ordering of the suites. Spades < Hearts < Diamond < Clubs. So, the first card that you would hand over to your partner would be of the same suite as the fifth card.

Now, we have narrowed down the possibilities to 12 cards ( as 1 card of the same suite has already been handed over to the partner ). It seems that now we have 3 cards left and 12 possibilities. How can we communicate a number upto 12 with only 3 cards? It turns out that we don’t need an ability to communicate a number up to 12. There is a circular ordering in the suite. so Ace < 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9 < 10 < Jack < Queen < King < Ace. By this scheme, the maximum distance between any two cards is 6. Now, we have 3 cards to communicate a number up to 6. well, 3! = 6. We can use permutation of remaining three cards to communicate this number.

But for , permutation we need a total ordering. We have already ordered the suites. And within the suite we will follow ordering of Ace < 2 .. < 10 < Jack < Queen < King ( We can’t use circular notion over here for obvious reason ). For example “King of Spades” is smaller than “2 of Hearts”. Remaining three cards now can be viewed in ascending order.

For example “King of Spade”(1) < “5 of Diamond” (2)< “7 of Diamond”(3). We can assign them numbers as indicated in the bracket. Now, we can assign number to the permutations :

123 = 1

132 = 2

213 = 3

231 = 4

312 = 5

321 = 6

Order in which you hand over the remaining 3 cards would indicate the minimum distance between the first card handed over and the card to be guessed ( the fifth card ). But how do we communicate the direction of the distance ? Counting up? or Counting down? Well, we don’t have to. The trick lies in picking up the card out of 2 cards of the same suite. For example “4 of Spade ” and “Queen of Spade”. You can tell your partner to always count upwards. Which card to hand over as the first card? If we hand over “4 of Spade” then distance in up direction is 7 ( as Queen = 11 ). So we would hand over the “Queen of Spade”. Queen < King < Ace < 1 < 2 < 3 < 4 = 6 hops ( or distance ).

To summarize, the first card should be chosen such that it has the same suite as the 5th card as well as distance in up direction should be at most 6.

Pretty neat, isn’t it?

–Saurabh Joshi

Tags: card puzzle, math, puzzle

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