Phil Anthropist likes to give away money. One day he went to the park with a wad of cash. He gave $100 to a man feeding pigeons. He then gave half of his remaining money to a child eating an ice-cream cone. He then gave $50 to the balloon seller. He then bought a hot dog and paid for it with a $20 bill. "Keep the change for a tip," he said to the hot-dog seller. Then he gave half his remaining money to someone giving a speech on a soapbox. At this point, he had $3 left and stuck it under the collar of a stray cat. How much money did Phil have when he started?

Let X be the amount Phil started with, in dollars. According to the problem, Phil first gave $100, then half of the remaining amount, then $50, and then the hot dog which cost him less than $20 because he gave the seller a tip. Let H be the cost of the hot dog in dollars, and T is Phil's tip to the hot-dog seller, then H+T=$20. After that, he gave half of his remaining money and then had $3 left.

We can write the equations below:
X - $100 - (X - $100)/2 - $50 - H - (X - $150 - H)/2 = $3

Combine like terms and simplify:
(2X - $300 - $150 - 2H)/2 = $3

Multiply both sides by 2 and simplify the equation:
2X - $450 - 2H = $6

Now we have:
2X - 2H = $456
H + T = $20

We know that T > 0 because a tip is given. Therefore, H < $20. We can test some H values:

If H = $19, we have:

2X - 2($19) = $456
X - $38 = $228
X = $266

In this case, T = 1, and this satisfies all conditions in the problem.

Thus, Phil had $266 when he started.

Well, it seems like Phil Anthropist had quite the adventure at the park. Let's do some clown math to figure out how much money he had when he started.

He started by giving $100 to a man feeding pigeons, so he had $100 less.

Then he gave half of his remaining money to a child with an ice-cream cone. Let's say he had X dollars left at this point. That means he gave away X/2 dollars, leaving him with X/2 dollars.

Next, he gave $50 to the balloon seller. So now he had X/2 - $50.

After buying the hot dog, he had X/2 - $50 - $20.

When he gave away half his remaining money to the soapbox speaker, he was left with (X/2 - $50 - $20)/2.

Finally, he had $3 left, which means he had (X/2 - $50 - $20)/2 = $3.

Using clown algebra, we can solve this equation to find X:

(X/2 - $50 - $20)/2 = $3.

We can simplify this equation and solve for X:

X/4 - $70/2 = $3,
X/4 - $35 = $3,
X/4 = $3 + $35,
X/4 = $38,
X = $38 x 4 = $152.

So, Phil Anthropist had $152 when he started his generous adventure in the park.

Let's break down the steps to calculate how much money Phil had when he started:

1. Phil gave $100 to a man feeding pigeons.
Money Remaining: $100

2. He then gave half of his remaining money to a child eating an ice-cream cone.
Money Remaining: $100/2 = $50

3. He gave $50 to the balloon seller.
Money Remaining: $50 - $50 = $0

4. Phil bought a hot dog and paid for it with a $20 bill, and told the hot dog seller to keep the change as a tip.
Money Remaining: $0 + $0 = $0

5. He then gave half of his remaining money ($0) to someone giving a speech on a soapbox.
Money Remaining: $0/2 = $0

6. Phil had $3 left and stuck it under the collar of a stray cat.
Money Remaining: $0 + $3 = $3

Therefore, Phil had $3 when he started.

To determine how much money Phil had when he started, we need to track his transactions step by step.

1. Phil gave $100 to a man feeding pigeons.
If we assume this was the first transaction, Phil had $100 left after this.

2. Phil gave half of his remaining money to a child eating an ice-cream cone.
Let's assume his remaining money was $x. He gave half of that, which means he gave away $x/2. So he had $x/2 left.

3. Phil gave $50 to the balloon seller.
After this transaction, Phil had $x/2 - $50 left.

4. Phil bought a hot dog and paid with a $20 bill, keeping the change as a tip.
If we assume the hot dog cost $y, then he had $x/2 - $50 - $20 + $y left.

5. Phil gave half of his remaining money to someone giving a speech on a soapbox.
Let's assume his remaining money (after the hot dog purchase) was $z. He gave away $z/2, so he had $x/2 - $50 - $20 + $y - $z/2 left.

6. Phil had $3 left and stuck it under the collar of a stray cat.
This means that $x/2 - $50 - $20 + $y - $z/2 = $3.

Now we can solve for the initial amount of money Phil had when he started by adding up all the transactions:
$100 + $x/2 - $50 - $20 + $y - $z/2 = $3.

Simplifying the equation:
$x/2 - $70 + $y - $z/2 = $3.

At this point, we need more information or additional equations to solve for the initial amount. Without that, we cannot determine how much money Phil had when he started.