Annie loves giving away puppies. When her friend Bobby walks by, she instantly gives him 50% of her puppy stash. Bobby, stunned with this shower of puppies, gives back 20% of his total. Annie counters by giving Bobby 50% of her new puppy total. Bobby responds by giving Annie back 50% of the puppies he now has and then running off, having gone from 0 to 7 puppies in just a few seconds! How many puppies did Annie begin with?

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To find out how many puppies Annie began with, let's break down the given information step by step:

1. Annie gives Bobby 50% of her puppy stash.
This means Annie is left with 50% of her initial puppies.

2. Bobby gives back 20% of his total.
Bobby now has 80% of the puppies he initially received from Annie.

3. Annie gives Bobby 50% of her new puppy total.
Bobby now has the puppies received from Annie plus the puppies he had after giving back 20%.

4. Bobby gives Annie back 50% of the puppies he now has.
This means Bobby is left with 50% of the puppies he had after receiving from Annie.

5. Bobby has 7 puppies at the end.
From the previous step, we know that Bobby had 50% of the puppies he had after receiving from Annie.

Now, let's solve the problem using algebra:

Let's assume Annie initially had x puppies.

1. Annie gives Bobby 50% of her puppy stash:
Annie has 0.5x puppies remaining.

2. Bobby gives back 20% of his total:
Bobby has 0.8 * 0.5x = 0.4x puppies.

3. Annie gives Bobby 50% of her new puppy total:
Bobby has 0.4x + 0.5 * 0.5x = 0.4x + 0.25x = 0.65x puppies.

4. Bobby gives Annie back 50% of the puppies he now has:
Bobby has 0.5 * 0.65x = 0.325x puppies remaining.

5. Bobby has 7 puppies at the end:
0.325x = 7.

Now, we can solve the equation to find x, the number of puppies Annie began with:

0.325x = 7
x = 7 / 0.325
x ≈ 21.54

Since the number of puppies cannot be fractional, Annie must have initially had 21 puppies.