Card Games: Concentration (again)

In the previous problem , we learned abouot the card game Concentration , otherwise known simply as the Memory Game .

If you are not familiar with the rules, they are explained in the previous problem.

This time, we will be using 142857 pairs of cards (this is known as the phoenix number). The phoenix number has these properties:

$142857 × 1 = 142857$

$142857 × 2 = 285714$

$142857 × 3 = 428571$

$142857 × 4 = 571428$

$142857 × 5 = 714285$

$142857 × 6 = 857142$

$142857 × 7 = 999999$

Back to the problem. Calculate the expected number of flips until two matching cards are seen (rounding to 7 decimal places is advised).

Extra Credit:

Generalise this for any $n$ pairs of cards.

More Problems About Card Games:

Concentration (Memory Game)

War

Play Your Cards Right