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Spot It problem, version 2

The game "Spot It!" consists of a deck of cards.

In the original version of the game, every card has 8 symbols on it. If you pick any two cards, exactly one symbol will be the same on the cards.

In the kids version of the game, every card has 6 symbols instead, but you still have exactly one match of symbols if you pick any two cards.

You may also assume that the deck contains the same number of every symbol (that is, no symbol is more common than another).

Suppose you want to create a more complex version of the game, where you have N symbols on each card and every two cards share exactly M symbols.

a) Is this possible?

In a deck with N symbols on each card, and each pair of cards share M symbols:

b) How many cards will there be at most?

c) How many symbols will there be in total?

(See also https://brilliant.org/discussions/thread/spot-it-problem-version-1/)

Note: The combination of symbols matching two cards should, if possible, be different for every two cards. (Thanks for Utkarsh Dwivedi for the question leading to this clarification.)

Note by Johan Falk
1 year, 11 months ago

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The problem is more complex than I thought....

Let us try to find all the solutions here..

First let me try to prove that for any value of N, there can be at least two values of M , namely 1 and {N-1}

Let there be D cards in the deck, each with N Symbols. Let any pair of cards have M symbols in common. Let the total symbols be S.

If there is any case of S = N, then two cards are enough!! To avoid the aforementioned contradiction, let us say that there is atleast one more symbol other than N [i.e] D > 2

Now, take the first card with N Symbols. We can have C [ N , M] more cards with suitable combination.

==> Max. No. of Cards D = C [ N , M] + 1 .................{1}

Now, Least Value of D is given by least value of C [ N,M ]

C [N,M] = 1 ==> M = {0, N} ==> D =2 ... Contradicts with our assumption..

C [N,M] = N ==> M = { 1, N-1 } ==> D = N+1 ...............................{2}

Hence, we have proved that for any value of N, atleast two values of M , i.e M = { 1, N-1 } are available

Again, we know that each card has [N-M] distinct symbols from first one...

Hence, No. of Symbols S = C [ D , N - M ] ...............................{3}

We can get S in another method also. The first card have N distinct Symbols. The second card has [N-M] distinct symbols. The third card has [N-2M] distinct symbols and so on....

Let Q {N,M} and R {N,M} be the Quotient and remainder respectively, when N is divided by M.

We know that A = B . Q {A,B} + R {A,B}

Now, Total No. of symbols S = N + [N-M] + [N-2M] + .... + [N - Q {N,M} M]

= N [1+1+..... {Q {N,M} + 1} times ] - M [1+2+...+ Q {N,M} ]

= N [ Q {N,M} +1 ] - M [ Q {N,M} ] [ Q {N,M} + 1 ] / 2 = [ Q {N,M} + 1 ] [ 2 N - M . Q {N,M} ] / 2

==> S = [ Q {N,M} + 1 ] [ N + R {N,M} ] / 2 .......................{4}

To find S for the cases M = { 1, N-1 }

M = 1 ==> S = N [ N+1 ] / 2
M = N-1 ==> S = N+1

Both methods give the same result. Hence, the result is the optimal most one.

To Conclude,

a) For any N symbols per card , there exists atleast two natural numbers M such that any pair of cards have exactly M symbols in common

M = { 1, N-1 }

b) No. of cards = N+1 , for both the cases

c) No. of Symbols

Atmost S = N [ N+1 ] / 2 for M = 1

The other result M = { N-1 } gives us [ N+1 ] only

Further findings:

Here are few more findings which I found during attempt to solution. But, I donot know how to prove them:

  1. For any prime No. P, only two solutions are possible M = { 1, P-1 }

  2. For any N, the only co prime for which a solution is available is M = N-1 {i.e} Any set of Co primes N , M do not have a solution unless M = N-1

  3. Non Co-prime sets N, M

If N = f A and M = f B, f being the H.C.F of N and M , then the problem should be rephrased as f { A , B } . Now, the problem is to be solved as above.

No. of Cards for { N , M } = No. of Cards for { A , B }

No. of Symbols for { N , M } = f * [ No. of Symbols for { A , B } ]

P.S: Thank you for the problem which occupied my day. I hope that I have answered correctly. Arun Palaniappan · 1 year, 11 months ago

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Can the same group of symbols be common to more than two cards ? Utkarsh Dwivedi · 1 year, 11 months ago

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@Utkarsh Dwivedi Good question. I would say no, and I should update the question to say that not only every symbol should be equally frequent – but also every combination of symbols between cards. (I'm not sure this is possible, though…)

Thanks! Johan Falk · 1 year, 11 months ago

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