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    <p>I just found a way to do it with 57 or 58 pictures but now I have a very bad headache, I'll post the ruby code in 8-10 hours after I slept well! just a hint my my solution every 7 cards share same mark and total 56 cards can be constructed using my solution.</p> <p>here is the code that generates all 57 cards that ypercube was talking about. it uses exactly 57 pictures, and sorry guy's I've written actual C++ code but knowing that <code>vector &lt;something&gt;</code> is an array containing values of type <code>something</code> it's easy to understand what this code does. and this code generates <code>P^2+P+1</code> cards using <code>P^2+P+1</code> pictures each containing <code>P+1</code> picture and sharing only 1 picture in common, for every prime P value. which means we can have 7 cards using 7 pictures each having 3 pictures(for p=2), 13 cards using 13 pictures(for p=3), 31 cards using 31 pictures(for p=5), 57 cards for 57 pictures(for p=7) and so on...</p> <pre><code>#include &lt;iostream&gt; #include &lt;vector&gt; using namespace std; vector &lt;vector&lt;int&gt; &gt; cards; void createcards(int p) { cards.resize(0); for (int i=0;i&lt;p;i++) { cards.resize(cards.size()+1); for(int j=0;j&lt;p;j++) { cards.back().push_back(i*p+j); } cards.back().push_back(p*p+1); } for (int i=0;i&lt;p;i++) { for(int j=0;j&lt;p;j++) { cards.resize(cards.size()+1); for(int k=0;k&lt;p;k++) { cards.back().push_back(k*p+(j+i*k)%p); } cards.back().push_back(p*p+2+i); } } cards.resize(cards.size()+1); for (int i=0;i&lt;p+1;i++) cards.back().push_back(p*p+1+i); } void checkCards() { cout &lt;&lt; "---------------------\n"; for(unsigned i=0;i&lt;cards.size();i++) { for(unsigned j=0;j&lt;cards[i].size();j++) { printf("%3d",cards[i][j]); } cout &lt;&lt; "\n"; } cout &lt;&lt; "---------------------\n"; for(unsigned i=0;i&lt;cards.size();i++) { for(unsigned j=i+1;j&lt;cards.size();j++) { int sim = 0; for(unsigned k=0;k&lt;cards[i].size();k++) for(unsigned l=0;l&lt;cards[j].size();l++) if (cards[i][k] == cards[j][l]) sim ++; if (sim != 1) cout &lt;&lt; "there is a problem between cards : " &lt;&lt; i &lt;&lt; " " &lt;&lt; j &lt;&lt; "\n"; } } } int main() { int p; for(cin &gt;&gt; p; p!=0;cin&gt;&gt; p) { createcards(p); checkCards(); } } </code></pre> <p>again sorry for the delayed code.</p>
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    1. POWhat are the mathematical/computational principles behind this game?
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    1. COI have an elegant proof of this, but alas this comment box is too small to contain it.
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    2. CO@Gajet: Did you run it for `p=4`? (and 21 cards/pictures)
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      1. USypercubeᵀᴹ
    3. CO4 isn't doesn't work in my algorithm since 4 is not a prime number, in my algorithm it's important that p should be prime.
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