[Guest]

There's a standard pack of 52 playing cards, arranged in the following manner:

1S, 3S, 7D, 5H, 12C, AC, 5S, 11H, 11C, 12D, 2D, 8C, 6S, 13H, 2H, 6D, 3D, 10S, 8D, 7C, 4S, 9D, 9C, 8H, 5D, 1D, 6H, AH, 4D, 12S, 12H, 13C, 3C, 9H, 7S, 2S, 5C, KD, 4H, 10C, 10D, JS, 10H, 9S, 6C, 2C, 7H, 3H, 8S, 4C, JD, KS

Problem 1:

Can you determine the rule which this arrangement follows?

Problem 2, More difficult:

You will notice that the arrangement above is cyclical so there is thus no natural "start point" to the sequence. Given a pack of cards that follows this arrangement, if one knows the top card, what is the formula to determine:

a) The position of any other card in the deck. For example, if the top card is 8H, at what position in the deck is 2C?

b) The card at any position in the deck. For example, if the top card is 8H, which card is in position 22?

I haven't actually solved this myself yet, but in theory, it seems that it should be possible to find such a formula...

[Guest]

You're not consistent. Does it matter that...

you're using 1S and 1D for two of the aces, but you're using AC and AH for the other two aces?

Also, 12C, 11H, 11C, 12D, 13H, 12S, 12H, 13C, KD, JS, JD, and KS.

In some you're using numerical values for the face cards but others you're not.

This isn't part of the puzzle is it?

Edited October 11, 2010 by enrightmcc

[Guest]

You're not consistent. Does it matter that...

you're using 1S and 1D for two of the aces, but you're using AC and AH for the other two aces?

Also, 12C, 11H, 11C, 12D, 13H, 12S, 12H, 13C, KD, JS, JD, and KS.

In some you're using numerical values for the face cards but others you're not.

This isn't part of the puzzle is it?

Sorry. My bad. I wanted to use numbers for values and letters for suits, but its natural to use letters for the picture cards so I must have gotten them mixed up when typing out the sequence. These are not part of the puzzle. Please ignore the mistake.

[bonanova]

Make a numbered array to visually inspect for patterns.

Rows are suits: Spades=1 Hearts=2 Diamonds=3 Clubs=4

Columns are rank: Ace=1, ... Jack=11, ... King=13

Values are index in the series.

For example, the 4th card in the sequence is the Heart 5 [suit=2 rank=5]

So array element [2,5] receives the value 4. Continuing,

1 36 2 21 7 13 35 49 44 18 42 30 52
28 15 48 39 4 27 47 24 34 43 8 31 14
26 11 17 29 25 16 3 19 22 41 51 10 38
6 46 33 50 37 45 20 12 23 40 9 5 32

[Guest]

1) You will notice that the arrangement above is cyclical so there is thus no natural "start point" to the sequence.

2) I haven't actually solved this myself yet, but in theory, it seems that it should be possible to find such a formula...

I'm sure if (1) is true, (2) is true

[Guest]

Make a numbered array to visually inspect for patterns.

Rows are suits: Spades=1 Hearts=2 Diamonds=3 Clubs=4

Columns are rank: Ace=1, ... Jack=11, ... King=13

Values are index in the series.

For example, the 4th card in the sequence is the Heart 5 [suit=2 rank=5]

So array element [2,5] receives the value 4. Continuing,

 1 36  2 21  7 13 35 49 44 18 42 30 52

28 15 48 39  4 27 47 24 34 43  8 31 14

26 11 17 29 25 16  3 19 22 41 51 10 38

 6 46 33 50 37 45 20 12 23 40  9  5 32

That's an interesting approach. In fact, I'm going to try and use it myself to try an solve part 2. However, for part 1, I'd suggest a simpler approach:

Try approaching it as you would any other number sequence/pattern on this forum. After all, each card is just a two-part number.

I found it easier once I ignored the suits altogether, and focused on just predicting the value of a card based on the other cards. Once I had this, it was easier to tackle what determined the suit.

Try assigning the suits the following numbers: S = 1, H = 2, C = 3, D = 4.

The suit is determined using two inputs. One of them is the value of the card in question.

The value of any card can be completely determined by the card preceding it. Both the value, and the “suit number” (from hint 3) are used, and a modulus of 13 may be required in some instances.

The suit of card n is determined by the value of card n, and the suit of card n-1.

[Guest]

1) You will notice that the arrangement above is cyclical so there is thus no natural "start point" to the sequence.

2) I haven't actually solved this myself yet, but in theory, it seems that it should be possible to find such a formula...

I'm sure if (1) is true, (2) is true

Yes, that's what I thought a couple of months ago. After a 100 failed attempts though, you begin to doubt everything...

Just a bit of background on this problem:

This actually is a very commonly used card arrangement - by advanced magicians. It looks completely random, yet once the arrangement is understood, some very simple calculations can provide the magician with a surprising amount of info about the deck.

The arrangement, in its current form, has been around for more than 25yrs in the magic community. The advantage of the arrangement relative to other arrangements is its mathematical base (so you don't have to memorize a random stack - which many magicians still do), but yet its completely innocent, random appearance (most mathematical stacks can be spotted in 2 mins by a 10 yr old). However, the one disadvantage of this stack relative to other mathematical arrangements, has been that there is no known answer for the two questions asked in part 2 of the OP (short of calculating each card, one by one, in one's head).

So two months ago, I decided I would serve all magician kind and bring some mathematical rigour to this popular arrangement. So far however, I've failed. Hopefully, the combined brain power of this forum might have better luck than I did...

[superprismatic]

Suit (S): Spade=0 Heart=1 Club=2 Diamond=3

Value (V): Ace=0 Deuce=1 Trey=2 Four=3 Five=4 Six=5 Seven=6 Eight=7 Nine=8 Ten=9 Jack=10 Queen=11 King=12

Function (F): F(x)=0 for x=0,1,2 ; F(x)=2 for x=3,4,5 ; F(x)=3 for x=6,7,8 ; F(x)=1 for x=9,10,11,12

Then:

V(n)=2*V(n-1)+S(n-1)+2 (mod 13)

and S(n)=F(V(n))+S(n-1) (mod 4)

And so,

[V(n),S(n)]=[2*V(n-1)+S(n-1)+2 (mod 13), {2*V(n-1)+S(n-1)+2 (mod 13)}+S(n-1) (mod 4)]

which is a function of [V(n),S(n)] in terms of [V(n-1),S(n-1)].

This is a mess because of the mixture of mod 13 and mod 4. Good luck getting a simple closed form

expression for [V(n),S(n)] in terms of [V(0),S(0)]! For a particular n, it's easy to do it with

something like Mathematica but the result is horrendously complex!

[Guest]

Suit (S): Spade=0 Heart=1 Club=2 Diamond=3

Value (V): Ace=0 Deuce=1 Trey=2 Four=3 Five=4 Six=5 Seven=6 Eight=7 Nine=8 Ten=9 Jack=10 Queen=11 King=12

Function (F): F(x)=0 for x=0,1,2 ; F(x)=2 for x=3,4,5 ; F(x)=3 for x=6,7,8 ; F(x)=1 for x=9,10,11,12

Then:

V(n)=2*V(n-1)+S(n-1)+2 (mod 13)

and S(n)=F(V(n))+S(n-1) (mod 4)

And so,

[V(n),S(n)]=[2*V(n-1)+S(n-1)+2 (mod 13), {2*V(n-1)+S(n-1)+2 (mod 13)}+S(n-1) (mod 4)]

which is a function of [V(n),S(n)] in terms of [V(n-1),S(n-1)].

This is a mess because of the mixture of mod 13 and mod 4. Good luck getting a simple closed form

expression for [V(n),S(n)] in terms of [V(0),S(0)]! For a particular n, it's easy to do it with

something like Mathematica but the result is horrendously complex!

Spot on! Well done. That’s the answer for part 1.

Yes, the part 2 does seem to be very complex. In addition to the mod functions, its the F(x) function you defined above that really makes this difficult to crack.

Edited October 13, 2010 by rajat_magic