A pack of $52$ playing cards is to be shuffled as follows:
I. Putting the top $7$ cards (one-by-one) on each other on a table. Lets call this group of the $7$ cards GROUP $A$.
II. Putting the top $3$ cards (one-by-one) on each other on the table. Lets call this group of the $3$ cards GROUP $B$.
III. Putting the top $3$ cards (without shuffling them) on the top of GROUP $B$.
IV. Putting the top $7$ cards (one-by-one) on the top of GROUP $A$.
V. Putting the top $3$ cards (one-by-one) on the top of GROUP $B$.
VI. Putting the top $3$ cards (without shuffling them) on the top of GROUP $B$.
VII. Repeating steps IV, V, and VI until all the cards are placed on the table.
VIII. Putting GROUP $A$ on the top of GROUP $B$.
Doing these $8$ steps completes $1$ run of shuffling.
How many runs do we need to make the $52$ playing cards having the same original order?
Any hint to solve this problem will be appreciated.
One approach would be to first write the deck shuffling as a permutation in two-line notation and then to translate this to cycle notation. With some work you can then reduce this to a product of disjoint cycles, and compute the least common multiple of the lengths of the cycles to find the number of shuffles required.