Pairs game
Task number: 4455
How many different starting positions does the famous Pairs game have?
In other words, how many ways can 64 cards containing 32 identical pairs be arranged in a square \(8 \times 8\)?
Solution
If we differentiate all cards, e.g. by numbers from 1 to 64, we would get \(64!\) possible (physical/numbered) placements.
In a single starting position, each of the 32 pairs of identical cards can be swapped without changing the starting position.
In other words, one unique starting position corresponds to \(2^{32}\) different (physical/numbered) placements.
We get the number we are looking for by dividing the number of all placements by the number of placements that correspond to a single starting position.
Answer
There are \(\frac{64!}{2^{32}}\doteq 2.95\,\cdot\,10^{79}\) different card layouts.
