Cunning Card Trick – A Solution 1^{st} May 2007
Solution provided by Alexander Gunasekera and James Currah, both year 5 pupils of The Batt Primary School, Witney
At the end of the trick, let the three chosen piles be called pile A, pile B and pile M.
The remaining cards are in the magic pile P.
pile A pile B pile M Magic Pile P
Pile A and Pile B are the two piles revealed at the end, pile M is the mystery pile. We know the values of A and B (eg ace is 1, 2 is 2 ... Jack is 11 and so on) as A and B are the values of the bottom cards revealed. We do not know the card M. We want to find M out.
Let the number of cards in pile A be nA.
Let the number of cards in pile B be nB.
Let the number of cards in pile M be nM.
Let the number of cards in pile P be nP.
We know what nP is because we can count the cards in the magic pile.
Take the pile A for example:
If the bottom card is 5 then there are 9 cards in the pile (5, 6, 7, 8, 9, 10, J, Q, K).
If the bottom card is J (11) then there are 3 cards in the pile ( J, Q, K).
Generally we can say that the number of cards in pile A (nA) is equal to 14 subtract the value of the bottom card, A.
Or: nA = 14 – A
Similarly, nB = 14 – B
And nM = 14 – M
We also know that there are 52 cards in a pack so we know:
nA + nB + nM + nP = 52
Using the information above:
14  A + 14 – B + 14 – M + nP =52
Tidying this up gives:
42 – A – B – M + nP = 52
Tidying again gives:
nP – A – B – M = 10
and again gives
M = nP – A – B – 10
Remember, M is the value of the mystery card we want to find.
In the trick, to find M we count the number of cards in the magic pile, subtract A cards then subtract B cards then subtract 10 cards. The number of cards left is M.
This works because we have shown that, M = nP – A – B – 10 above.
